Current Build (Level 28, not finished and not with entirely optimal runes, but used for the purpose of the calculations): AD/level Marks, AD/level Quints, Mana Regen/Level Seals, CDR/level Glyphs.

Masteries: Everything in Offense Apart from Mental Force, Butcher, Demolitionist, Deadliness, Lethality (who comes up with these names?), Vampirism, Sunder, Archmage and the last point in Blast.

Last 11 points used for tests have been in utility for mana purposes and red buff duration extention, but 11 in defence is likely better lategame and will be tried next time I play this build.

Item build: Merc threads, Gunblade, Nashor's Tooth, Lichbane, Rabadons Deathcap, Rageblade

536.5 ap before rageblade proc, 599.3 with. 205 attack damage, 1.729 attack speed before passive and rageblade, 2.549 attack speed after passive and rageblade. ~40% cdr (11~% from runes, 4% from masteries, 25 % from nashors).

To make things easier, I will use pre-rageblade AP values on abilities and such.


Mystic Shot: 115 + 205 + (536,5 * 0.2) = 427,3 Physical Damage. 2.4 CD, 1.4 CD if hits.

Essence Flux: 280 + 375.2(AP) = 655.2 Magic Damage. 5.4 CD.

CD if 2 mystic shots hit while it is on cooldown: 3.4

Arcane shift: 275 + 402.4 = 677.375 Magic damage. 6.6 CD.

CD if 2 mystic shots hit while it is on cooldown: 4.6

Trueshot Barrage: 650 + 205(AD) + 482.85(AD) = 1337.85 Magic Damage

Lichbane Proc: 536 Physical Damage.

Gunblade Active: 300 Magical Damage.


Initial Combo against 1 person if ultimate hits (Gunblade slow makes dodging difficult):

1337.85 + 677.375 + 655.2 + 300 Magic Damage. 205(auto attack) + 427,3 + 536 Physical Damage.

2970.425  Magic Damage + 1168.3 Physical Damage before resistances.


Ezreal's low cooldowns in this build and mystic shot passive makes 2970.425 – (1337.85 + 300) =  1332.575 Magic damage (Essence Flux and Arcane Shift) come up again very quickly.

On top of this comes 205 * 2.5 = 500 physical dps from auto attacks if he can dare stay in range, and there is mystic shot at 427.3 every 1.4 seconds as well as lichbane procs in at 536.5 – 599.3 damage somewhere between every 2.8 seconds and 2.0 seconds(including rageblade and passive here. Passive gets filled instantly or near-instantly, rageblade gets half-filled instantly, then filled quickly).

I now realize I have forgotten to include Havoc and Executioner in calculations, but I feel too lazy to do them again.

3 of Ezreals damage sources have a potential 900 range or more if he needs to be extra careful (Mystic Shot + Lichbane proc and Essence Flux). Arcane shift has a 750 Range with a teleport-in, making it a potentially good finisher or something to use to hop in range of 1 or more persons and burst them with Mystic Shot, Lichbane proc and Essence Flux (after ulting them from out of sight or when they are engaged/it is difficult for them to move).


How to start out build?
Maxing Q to be able to get in a second hit in a potential full on fight, and to synergize with your best first item. Bilgewater Cutlass give you AD, sustain, a slow to help ganks or to make kills easier, and 150 magic damage of burst. Turn into Gunblade for more burst (now also from AP) and higher range of active. The active slow can aid with ult-hitting. After Gunblade, build Sheen to further rely on your high burst and your low Q cooldown by midgame. At this point, I am not completely sure what is best.

I don't know what the best build progression will usually be. I see different paths that require adoption of slightly different midgame/late midgame playstyles:

Nashors-->Lichbane.
Requires you to be rather careful as your burst falls off when you build Nashors instead of something else. Your ultimate will be up more often and your abilities will too. You also get a boost to auto attack speed. You get as much mana regeneration as you will need to be comfortable.

Rabadons--> Lichbane
You're a hurr durr daumn mage and don't you forget it. Continues to strengthen burst, but makes you very ultimate-dependent before lichbane. Sustained damage not the best.

Lichbane--> Rabadons
Relying on Q a lot. Your AP isn't that high yet before rabadons, but you will double the damage of sheen. Provides  a high mana pool and defense against evil leblancs and ahris quite early, together with your mercs. Probably the worst choice. I have not tried it yet. I doubt I will.

Rageblade--> Lichbane
Decent burst and auto attack damage. Your Arcane shift is still on a relatively high cooldown as you aren't level 18 yet, and you level it last, so you don't gain as much damage from AP as due to its high cooldown, you need to be very careful about when to use it. (You want to partially shut down down physical damage dealers and buff your own with max W at midgame/late midgame. The strength of 40% increased/decreased attack speed on 1-2 of your melee Ads and 1-2 of the enemy melee Ads, or on the enemy ranged carry in early teamfights is not to be underestimated). Your auto attacks will still remain useful.

Rageblade --> Fiendish Codex->Lichbane
At least for teammfights. decent derps and mobility. Does not add nearly as quickly to burst as other choices, but your ultimate still hits very hard. You get extra mana regeneration quite early in the phase where teams dance next to each other trying to engage in a favourable way, allowing to poke all day and use more Q  to speed up farming. You are very hybrid. Your burst can't take a squishy from 100 to 0 before lichbane unless you're ahead.

Rageblade --> Stinger ->Lichbane
Same as above, but maxes attack speed earlier at the expense of some burst and mana regeneration.

New thoughts: Bilgewater Cutlass -> Pickaxe ->Sheen -> Rageblade -> Stinger -> Rabadons -> Lichbane -> Gunblade

Building your early and midgame around Q and auto attacks, transitioning to Hybrid lategame.

Conclusions

Hybrid Ezreal is the most tricky champion to play of any champion with any build I have played so  far. I'm very far from mastering him, but my experience has been that I could always be useful to the point of carrying when I played somewhat well. He spams like a Skarner, if not more, but you actually have to think as quickly as you mash your buttons.

Pros

Range: Being able to deal a decent portion of his damage if he has to stay out of auto attack and/or gap closer ranges, and having a very low cooldown on the relevant abilities.

Arcane shift: As always, Ezreal can dodge a skillshot and still deal damage with this if he is careful and reacts quickly, or can simply get away or over a wall or something. Or kite. The build provides a very low cooldown. Mystic Shot also on a low cooldown helps this due to the mystic shot passive.

Sustained damage: Auto attack this is mainly a sidedish, albeit a useful one. If you are not being focused, your auto attack dps is not a negligible addition to the damage of your abilities.

The spellvamp + lifesteal helps him being topped off towards the end of a teamfight so long as he avoids being focused for too long or by too many people. If you're very low health, you can rather safely stand at high range and use Mystic Shot and Essence Flux until you have more health again. If it's at the start of a teamfight that you have low health, your intitial combo of your ultimate hitting followed by Arcane Shift, Mystic Shot with Lichbane proc and Essence Flux will heal you substantially.

Essence Flux Buff/Debuff

Cons:

Squishy as derp. The MR helps against being gibbed by a friendly mage, and the tenacity helps him to be able to flash out more quickly if he gets caught and, but other than that, any mistake of positioning can and likely will be fatal.

His 550 range makes it sometimes difficult to use this fully, though it is not any harder than with AD Ezreal, Graves, Vayne or Corki.

If you engage a carry lategame, you need to have everything except your ult up and maybe gunblade active to kill it first.

You're not ryze or a rightclick superhero. All your stuff is tricky to hit with. This makes the build one of the easiest builds to fail with if you do not play well. If you can't hit your skillshots, you become near-useless.

You're voulnerable to assasins. You don't have any hard CC.

Rune-specific: The current build does not use any flat AD or AP runes (I do not own these), making him weaker at the first levels compared to what he could be.