FLAME MECHANICS

(OUTDATED AS OF JUNGLE INFERNO UPDATE)

i apologize in advance that this is such an incoherent mess. i honestly dont have a very firm understanding of this stuff myself.
proceed with caution.

flamethrower is a projectile weapon. when you hold mouse1 it fires lots of little invisible boxes (particles). 1 flamethrower ammo fires two boxes. you can tap to shoot one box at a time for 0.5 ammo. the flame effect you see when you use press mouse1 is not a good indicator of how the particles really move.
when a particle makes contact with an enemy player's collision hull (hitbox), a straight line is drawn from the point of firing (your pov at the time of firing the particle) to the center of the particle that made contact. if there are no map brushes or solid entities in between the two points, the hit is confirmed and fire damage is applied.

make a local server and type this in the developer console:
sv_cheats 1
tf_debug_flamethrower 1
this reveals the true flame hitboxes and some other flamethrower data

its helps to imagine flame movement in two parts. inherent and inherited.
inherent movement can be observed by standing completely still and firing in any direction.
i'll leave the more detailed observations to you but keep these inherent properties in mind:
-each particle fired has a randomly generated (with parameters) speed and spread.
-"forward" velocity for each particle decreases over time.
-each particle accelerates upwards slightly near the end of its lifetime

when you are moving, your velocity at the time of firing a particle is added to that particle's inherent movement in the direction that you are moving. this is inherited movement.
for example: strafing left while firing straight up will send a stream of flame into the air as normal, but the stream will move left with you at 300hu/s (default pyro speed)
when you stop strafing left, any particles fired before you stopped moving, will continue traveling left regardless of your following movements.
likewise, if you strafe forward while firing straight forward, the stream will move along with you appearing to have no difference in range, meaning the particles are in fact moving an additional 300hu/s to inherent speed.
when you come to a stop, the last particles that were fired before stopping will travel noticeably further than any following particles. moving forward increases forward particle velocity, therefore increasing total range.
strafing backwards while firing forwards will add 270hu/s backwards (negative) velocity to inherent movement. this will result in a particle's initial forward velocity being slower.
as inherent forward velocity decreases, inherited velocity begins to catch up and exceed it; the particle will come to a complete stop before accelerating in the opposite direction like a boomerang.

now that we understand this principle we can begin to apply it in clever ways.

imagine there is a death pit between you and an enemy. you have no secondary ammo
you stand at the very edge of the cliff and fire your flamethrower to no avail; the flame just barely fails to reach him
he taunts you with "the schadenfreude" that he bought from the mannco store with his mother's credit card
taking advantage of this moment of vulnerability, you begin shuffling back and forwards while firing your flamethrower
your enemy, still in the middle of his taunt begins to feel horror and regret as he is bombarded with unexpected bursts of flame
he types "wtf" in the chat box before dying of afterburn as you initiate your own taunt
you look to the skies and thank sketchek for bestowing you with such fiery wisdom
~based on a true story

short bursts of flame while strafing forward can be surprisingly useful, not just for reaching otherwise unreachable targets, but it can help with spacing and aiming.
the longer range allows you to keep more distance between you and your enemies and the increased speed makes it easier to lead the enemy.

in situations where you are strafing left or right while flaming, such as circle strafing, you need to aim slightly away from the enemy (in the opposite direction you strafe) rather than lock onto them with the crosshair.
the more distance between you, the further away you need to aim. there are a few other situation you may want to side strafe and flame during fights, combos and ambushes but they are relatively rare and situational so you can think about it yourself if you really care.

strafing backwards while flaming is perhaps one of the most effective tactics in a pyro's arsenal. to demonstrate why, lets imagine a flamethrower fight between two pyros.
a red pyro chases a backpedaling blu pyro. both have their crosshairs locked on eachother and the red pyro is starting to catch up (backpedal speed is slightly slower than strafing forward)
view this gif for a rough simulation http://i.imgur.com/gLRYOfh.gif
even though the red pyro has faster particles and faster strafe speed, the range of the flame relative to the shooter is greater for the blu pyro.
the red pyro's inherited speed makes no difference if they both move in the same direction.
on the other hand, the blu pyro's particles may move slowly and float backwards after a moment, but the simple fact that they don't travel in the same direction as the pyros (at least not as fast as the chaser) means the red pyro will end up walking into them.
in addition to this, the one being chased is the one that chooses the path that both parties travel making it easier to lead the chaser and harder vice versa.
backpedaling around a corner and leaving a trail of flame for the chaser to walk into, makes it impossible for the chaser to do damage (excluding explosions) while the backpedaler does damage without even seeing the chaser.
flame hit detection draws a line from the point of firing to the point of collision so even if the enemy is unseen at the actual time of firing or time of contact, the hit is valid.
this is why you should almost never chase a backpedaling pyro, especially around corners.

due to particles inherently rising upwards over time, total range will generally be longer when aiming upwards than downwards. jump while flaming up for huge upwards range or down while falling to supplement the limited downwards range.
flaming while jumping or falling long distances can be difficult to aim because gravity isnt a constant value like max strafe speed, but once you understand this, its just a matter of adapting

these are just some of the many ways to use inherited movement to your benefit in a firefight.
when you understand how flame particles truly behave and how to properly aim and manipulate them, a whole new metagame opens up for you
your pyro game gains a new layer of depth and your success rate definitely increases to some degree.
i highly recommend everyone to become familiar with inherited flame movement even if you dont play pyro, as it could also help you fight against enemy pyros

FIRE CRITTING

(OUTDATED AS OF TOUGH BREAK UPDATE)

the following text details some of the ways to land burning criticals on players with afterburn immunity.
there are a lot more crit tricks that i won't bother going into due to them being insignificant or near impossible.
please know that even in the rare occasion that you succeed, most of these tricks are hardly ever useful in a competitive environment.
i recommend you practice the motions on a local server with bots before trying on live players. tr_walkway can be convenient but try proper maps too.

before we begin, know that there are two types of fire damage. "burn" and "afterburn".
burn occurs on contact with the damage source. damage is applied once and the "on fire" status induced by burn always lasts less than one second.
afterburn is applied immediately AFTER "BURN" occurs and does multiple hits of fixed damage over time. the fire can go on for several seconds.
pyros and targe demos (and now spycicle spies) are completely immune to afterburn and will only suffer burn damage.
this means the window to land burning critical hits is significantly smaller and may require specific techniques and conditions to be possible.
even if a particle makes contact at the very end of its lifetime, the burn time will still be the same, giving you maximum time before the crit window.
protip: heal beams shorten burn time, not just afterburn time. ie pyros with medic/dispenser/cart are very hard to crit.
the following text refers to "target(s)" as afterburn-immune players and "crit(s)" as burning crits or minicrits on said targets (unless specified otherwise ^_^)

part1. crit flaring(degreaser)
the easiest method to do this is by aiming at your target, holding mouse1 with the degreaser then switching to secondary, still holding mouse1 until the flare fires.
its as simple as that in terms of execution but when you bring all the other factors into account it gets a bit more complicated.
against a live opponent, you need to predict the target's actions while compensating with your own aim, movement and timing so that both the flame and the flare will make contact at the right timing without putting yourself in too much danger.
make sure youre very comfortable with inherited flame movement so you can hit flames from a variety of angles and situations.
depending on the time it takes for the flame to reach the target, you may need to adjust your flare shot timing to some extent or it could make contact too early or too late for the crit.
this doesnt necessarily mean releasing mouse1 after shooting your flames. you could try strafing further or closer, crouching or jumping, anything that changes the timing
i found that one of the safest and most effective ways to hit crit flares on pyros was by strafing forward for the flame burst, then backpedaling for the flare.
doing this allows you to keep distance, aim the flame accurately and time the flare perfectly for long distances.
you could airblast the target into the air to restrict movement, possibly making the shot easier. you would need enough height (2-3 airblasts~) to compensate for flame delay after airblasts.
note: crit flares can do significant knockback on lighter classes like pyros. you can bounce them into the air by aiming the flare at their feet or accelerate a fall by aiming for their head as they fall.

there are also airblast variations of the crit flare
if you are very close to your target, you can flame, airblast while moving in close and switch to flare immediately to crit right after the airblast.
if the target is hugging a wall (setup with airblasts) you can start from slightly further away but you need to move in and maybe jump so the flare shot is as close as possible and the airblast doesnt push them too far away.
switch speed and distance is the key here. the further away you are, the faster you need to switch. using a script can help.
this method can launch players quite some distance/height (aim flare at the feet for more distance)

the second variation is a bit trickier.
you need to airblast your target into flames rather than have the flame make contact before the airblast.
imagine a target is charging at you head on. you sidestep to the right, your crosshair also slightly off to his left as he comes within flamethrower range.
you then strafe forward-left so your crosshair lines up with the target at about mid-flamethrower range. the flame particles "curve" around and behind him allowing you to airblast him into the flames and crit flare him from outside of flamethrower range.
the critical knockback combined with the airblast sends him flying way out of range as you escape.
by having the flames hit the target after the airblast, you can stay safer for longer and take your time with your flare (although the specific timing may vary more) compared to the other variation.
you could also jump towards a target and flame as you fall down so the flames go over and behind him.
there are some situations where this method is possible without manually "curving" the flames around
for example you could airblast them high into the air against a wall and as they descends you puff some flame above him and quickly airblast him back up into the flames.
you can do this more consistently with no need for a wall by looking straight up and walking forwards underneath them while you flame to set up a chain of flames directly above the target.

you can also use the volcano fragment to ignite before switching to secondary for the crit. the timing for this is difficult to master.

its possible to chain a crit flare off a reflected flare if you walk forwards and switch really damn fast

these methods all apply to the detonator and scorch shot's minicrits but the knockback differs between each weapon e.g detonator wont bounce players off the ground.
ps you can apply fire to a target as a scorch shot dud explodes and it will mini crit. with good timing and positioning, you could get a long chain of mini crit scorch shots and duds.

part2: crit axing (degreaser)
this is where things start getting funky
you see, crit flares are simple because the flare is fired immediately as the weapon is deployed whereas the axtinguisher has a delay from swinging the axe.
flaming from point blank will easily crit with flares but will never crit with the axtinguisher.
the problem here is not the axe swinging too slow or the fire going out too early, the flame simply makes contact too early.
turn on flamethrower debug to view flame hitboxes and try flaming and axing some empty space in quick succession. youll notice that the axe finishes swinging before the particles die.
so the solution is to have the flame make contact with the target late enough so that the burn will be active at the time of the axe hit.
here are some tips on how to go about this

probably the most straightforward and consistent way to do this is by flaming an enemy as you descend from high above and axing as you come into melee range.
i don't have a very methodical approach to this technique, mostly relying on "feeling" gained through practice.
regardless, i find it helps to keep the following in mind:
flaming while falling will cause your flames to inherit your downward velocity at the time of firing resulting in more speed and therefore range.
however inherent flame properties (deceleration and ending rise) cause your flames to slow down.
meanwhile, gravity is still acting upon you, allowing you to soon catch up with your flames.
assuming you are falling from a suitable height, or moving at a sufficient speed, you can land a crit axe by switching to axe at the right time.
the switch timing is largely dependent on your speed, and partly your aim/the enemy's movements.
for example, if the flame makes contact with the top of the target's hitbox, you would need the axe slightly earlier than if it had made contact with the side of the hitbox.
airstrafing comes into both aiming flames and positioning for the axe.
this method allows some slight leeway with the timing but its usually more consistent to just fire a short puff and keep holding mouse1 while switching to axe so it swings as soon as possible
this way the only timing you really need to worry about is the switch to axe. think of it as market gardening but with a slightly longer melee swing time and its okay to touch the ground before the hit lands.
explosive jumps might be necessary for extra height if your starting position is too low.

its possible to use flare type weapons to ignite the target for the crit but requires much more speed/altitude than flames.
if you detonate on yourself immediately after a reflect jump, you will have enough height (and reload time) to use a flare for ignition on targets of level ground or lower.

some other ways to jump/drop axe in brief:

fall towards target, look straight up (maybe slightly towards the target) and flame just as you are about to land, switch to axe while strafing into melee position and swing as the flames arch downwards into the target.
this method allows you to axe targets from lower altitudes with some freedom of movement before the crit but is very hard to connect the flames
its possible to "stairstab" using this principle but the next method is better

flame downwards during a jump, or around the peak, causing the flames to rise upwards (sometimes through the ground) or travel slower/shorter
this method allows for very low altitude axe crits. this is the easiest way to stairstab pyros as pyro. the steeper the slope, the lower you aim and vice versa.
in fact, if the target runs straight at you, you could minicrit them from level ground.
if the slope is steep enough you can flame straight down while jumping backwards and succeed. "flipstabs" are also possible.
with detonator/scorch shot jumps, its possible to full crit from level ground
see: https://www.youtube.com/watch?v=x4SsS_MQHmk
you can combine this strategy with an airbast from above to stall the target in the air and still land the crit or possibly a minicrit that smashes them into the ground for fall damage

as we discussed earlier, backpedaling while flaming produces slower particles, pulling the particles back over time.
this means that if you suddenly started strafing forward after backpedal flaming, you would be very close to the particle as its lifetime ends.
if your target makes contact with a lingering particle at that point, the crit is yours.
the hard part is getting that to happen. while i suppose its technically possible from any position, the best way to set this up is similar to a cornerstab.
have the target chase you and right after turning a corner, fire some flames around where you predict they will follow you (easier in narrow areas like vents), then immediately strafe forwards and swing your axe.
if they were stupid enough to follow you and you got the timing right, the axe should minicrit right in their face as they come around the corner. with some tricky movement and a lot of luck you could even land the full crit.
you could forget the corner and just do this face to face but that would add a lot more risk. "matador" axe crits in the back are possible but highly improbable.
if youre feeling particularly evil you could lure an enemy player for a partner taunt and then cancel right when they taunt so they are immobilized and axe them.
if your target is still, you need to start aiming away and strafe backwards diagonally to time/position the particles.

if your target is falling down from up high near you, you can puff flames along their trajectory and then land a crit or minicrit. depending on how fast they are moving you may have no need for any fancy flaming.
you can still improve your chances by strafing backwards a bit as you puff your flames. you want the particles to stall above the area they will land. the faster they fall, the higher you should aim.
set this up with airblasts against high walls.
you could also stand directly below them, jump and flame up as you land, then jump and swing with axe. this isnt the safest or most consistent way to do this but it is possible.

its actually possible to flame>airblast>axecrit much like the curving flare crit variation but i never figured out how to make it work consistently.
its probably impossible without a wall or something to close the distance for the melee.

it is never possible to land a mini/crit axe just by strafing forwards unless both parties strafe directly into eachother with speed boost (whip,concheror etc) or its a targe demo charging at you.

note: the reserve shooter can allow faster or easier axe crits for some techniques, but does not open up any completely new/unique possibilities as far as i know.

it IS possible to use your secondary in between degreaser and axe and still land the crit but the timing, precision and luck needed is ridiculously strict (random speed/spread on inherent flame), even putting aside the unrealistic conditions needed for potential applications.
ive managed to get flame>detjump>axecrit and flame>flarecrit>axecrit on bots before, so i'll bet you could do a whole bunch of other crazy shit

almost all the techniques for degreaser axe crits can apply to the stock flamethrower+flares except the timing is later, stricter and the flareguns are projectile weapons. also the flaregun can bounce targets off the ground but the axt usually cant.

scorch shot duds stay in the world forever until they touch a surface and explode into fire, meaning you should have more than enough time to switch to axtinguisher from stock flamethrower.
you can set this up by airblasting the target up high against a wall, scorchshotting them from directly beneath so the dud doesnt hit the wall, one more airblast as they descend for timing and positioning, then axe as the dud hits the ground.
ideally, the target should be just about to land as the dud explodes.
..or you could just use the degreaser. one airblast against almost any wall, scorch shot to the body, axe. the timing is so easy you dont even need to think about it (but it does help to keep an eye on the dud just in case)

you could also crit axe pyros without degreaser if you shoot a flare from far away enough to swing your axe by the time it hits and catch up somehow. super long/fast jump/fall, teleport etc.

VARIOUS TIPS
ping/latency is a big deal when it comes to advanced flamethrowing. you want as little lag as possible. you also might want to consider practicing with artificial ping on local servers (net_fakelag).
there are some quirks regarding latency and the position of the flamethrower's firing point. look up "flamethrower mojo" for more info.
if you switch weapons away and back to degreaser during airblast cooldown, the cooldown lasts slightly longer.
airblast knockback is calculated relative to the position of your pov to the enemy. aim makes no difference. jumping and crouching will make differences in the height the target flies.
strafing against airblast will stop horizontal knockback but not vertical. this can help the pyro if he wants to keep airblasting you into the air
airblasting targets for fall damage height usually takes 3 airblasts. the first will either pin them to a wall so you can close distance and then use 2 crouched airblast or just 3 consecutive while they strafe against it unless they airstrafe away or something.
interestingly, pomson and bison projectiles use the same hitboxes as flames and can be viewed using the flamethrower debug mode. this explains why they light huntsman arrows.

..and thats all im telling you. wouldn't want to take away the fun of figuring stuff out yourself.
though hopefully i saved you some time by shedding a little light on the basics. now its your turn to come up with stupid ways to kill pyros.

THE END
i didn't cover absolutely everything i know but i've probably provided all the tools anyone would need to figure out the rest.
just remember that there isn't really such thing as a "pyro combo" in the same sense that there is a "backstab" in tf2.
there is no "burning drop axe crit combo", only the fact that you acted in a way that resulted in a critical axe on a pyro.
don't restrict yourself to arbitrary simplification. the whole purpose of playing this way is to be unconventional and creative.
~be fluid~