that dumb image

Q1: Why did the toot create synths if robots already exist?

A1: The toot was experiencing a power shortage, and didn't want to continue repairing and charging their army of gen1/2 synths. They were still robots, and ran on power sources and required mechanical repair. We learn that there are squads of gen1/2s that roam around the commonwealth cannibalizing settlements for parts, because their self-maintenance routine dictates they need those parts. Lots of people die from this, according to rumors (university point is not one such case though, the roaming gen1/2 squads happen off screen).
Gen3s were FEV derived beings, and thus had many of the benefits of FEV. They had tougher immune systems, they aged slower, they didn't need to eat as much, they don't need to sleep as much. FEV makes them "superhuman" essentially. They are the perfect labor force, from a practical point of view. A sleepless labor force that are strong and never get sick or age. On top of this, they have big human-like brains with which to problem solve in real time on the job.

Q2: Why did the toot make slaves that resented being slaves?

A2: They didn't. Their intent was to make slaves that had big human-like brains so they could be very autonomous in carrying out tasks. Simply telling a group of synths to mine a tunnel and figure it out is a lot easier than programming a gen1 synth how to do it from the ground up and creating whole new tasks in the programming. It was an unforeseen "bug" in the gen3 synths that they had things like hopes and dreams and desires. It's simply a side effect of having these big brains humans have. We will always wonder what is outside our walls, and what is out in the world. It's just natural. All the issues with gen3 synths are from having these brains that cause these thoughts.

Q3: Why did the toot create obedient synths that hunt other synths (coursers)? Why not just make all synths obedient?

A3: They didn't do this either. Coursers are chosen from among the general gen3 synth population based on personality traits. They are selected due to traits like loyalty, combat proficiency, ruthlessness, and determination. They are selected from among the general population and given further training, particularly combat and subterfuge training, and then graduate to be Coursers.

Q4: Why do synths have a failsafe that erases their memories of the toot when they escape, but this exact same technology not just shut them down when they escape?

A4: Synths escaping is fairly recent. Synths barely trickled out of the toot before Binet began helping them escape. Synths escaping only became a serious problem after Binet (Patriot) began his work of actually helping them. So the whole anti-synth escape programming is fairly new, and it might be a delicate procedure to start writing new prgrams to wipe this or that or shut down this or that when a synth escapes. But already having the program to wipe their memory when they find themselves on the surface makes sense, since that would be a priority even before they regularly escaped. They sent early model gen3 synths to surface to test them long before they were at war with the surface. So the memory wipe program makes sense to have day1 since test synths were on the surface, but intentional escapees only any noteworthy frequency have only been an issue in the last few years since young Binet started working.

Q5: Why does the toot not guard the teleporter?

A5: Because no one can invade that way. Nobody on the surface is remotely capable of entering the toot through the teleporter. It is only the player character, with the help of Father, and then the help of Virgil, and then the help of a faction that gains this ability. So of course there is no need to guard it. it took killing a courser, knowing about the chip, decoding the chip, virgil's help, sturges/Tom's help, Father allowing it to happen, and a good bit of luck for the plan to work.
And of course, when you get to the toot the first time, Father is expecting you and wants it to be welcoming. There could have been some gen2 guards there at most times but not then, perhaps. Remember, there are actually gen2 gaurds at many of the doors and division entrances even while you're there, so they do guard stuff, just not the thing that people can't enter without Father's permission (which you had when you entered the first time). He could have stopped you at any time by simply killing Virgil, killing Tom/Sturges, or by shutting off the teleporter when you were in it killing you. The entire teleporter plan only progresses with Father's permission, so of course it's not guarded.
Then later, the BoS blow a hole in the roof and the MM+RR lie and pretend you're still with the toot to enter.

Q6: Why did the toot need Sean if the toot is underground and shielded?

A6: It's not. Everyone in the toot is still mildly affected by radiation and very mildly mutated. They say this. They need Sean because he's actually unaffected by radiation, being prewar and in a pod. Even nate+nora are questionable, sicne they had pre-war radiation exposure from living in the unsafe nuclear age of nuclear powered cars and such. But Kellogg does call you the "backup", so I guess it was close enough.

Q7: Why didn't the toot put Sean in a rad suit during his trip to the toot

A7: It doesn't happen that fast, and the toot scientist did cradle him in her rad suit. And the Sean we see in Diamond City was already the synth version, as Sean was like 50+ years old at that point.

Q8: Why did the toot kill everyone in Vault 11?

A8: They weren't interested in saving them, they took what they needed and tied up loose ends. The toot is not benevolent. Even pre-war adults had some radiation exposure from daily life in the city, so it's possible baby Sean was truly the only thing they wanted and the adults were worthless. Kellogg is pretty quick to blast the spouse and doesn't seem to face any repercussions from that, which supports this.

Q9: Why did the toot allow the protagonist to run around the Commonwealth instead of taking them with them, or killing them, or keeping them frozen, or any number of things?

A9: It was a test. Sean wanted to see what you were like. Remember, when you get to the toot the first time, he says "I didn't expect you to survive out here, in all of this". He didn't know if you were going to even live. But if you did, he wanted you with all that experience to lead the Toot after his death. It was all to have someone new, some fresh blood, leading the toot. This is all explained. It was a neat little science experiment for him, seeing if you would live and get to him. He orders Kellogg to be where he is so you can kill him, he leaves Virgil and the RR and the MM all alive despite knowing where they are at all times 9yes even Virgil) so you can follow the breadcrumbs to him.

Q10: Why did Kellogg kill the spouse, if his own backstory was tragic about his wife being murdered?

A10: He was well over 100 years old at this point, his brain could have been damaged. on top of that, he was always pretty evil. Remember, even before his own wife was killed, he accepted jobs murdering innocent families over land disputes. There are also people who experience tragedy and are changed for the worse, and become resentful and spread their misery. He could have simply been one of those people.

Q11: Why does Kellogg refer to "the old man" and "father" even while kidnapping baby Sean?

A11: He doesn't. He says this in neither the initial kidnapping nor the memory replay later in the game.

Q12: What do supermutants in the commonwealth want?

A12: They want to establish their own state and kill all humans in the area so they rule the area. This is what they want. If you mean why are they here at all, it's because the Toot put them there. The toot made super mutants as part of their FEV research, and when they were done with them, they found it convenient to dump them on the surface to murder those pesky surfacers and keep them occupied so they don't go looking for the toot.

Q13: Why do supermutants kidnap people?

A13: To eat them.

Q14: Why do supermutants have bottle caps, and what and how would they even use them?

A14: Some small portions of a person's life are retained through the transformation. Not details, but vague feelings. This is why there is a supermutant who still loves his wife in Fallout1, and she loves him back. So it stands to reason that a vague sense of value may still be associated with bottlecaps even after transforming. They may collect them and not even remember why. They have no use for them, but they have a very vague sense of these being important, from their past life, so they collect them without knowing why. A postman with alzheimers feeling the desire to have a stamp collection without any memory of being a psotman, for example.

Q15: Why are there so many supermutants if they can't reproduce and they don't have any FEV vats to make more even with kidnapped people?

A15: the toot continued to make hundreds of mutants during the FEV experimentation. They kidnapped hundreds of people over the course of years, and then continued to make more mutants past the length of the experimentation because they didn't care and saw it as a way to attack the surface without having to do it themselves. This explains why they didn't just euthanize the experimentation mutants in the vat, they wanted them for the secondary purpose of destabilizing the surface as well.

Q16: Why does Virigl want to be human again if the toot will be looking for him?

A16: For one, they always knew where he was since day 1. They allowed him to escape and allowed the SS to find him because Father was watching you follow the breadcrumbs. X6-88 mentions that they've always known where he was, and we even know the how: watcher crows. But as for why he wants to transition back to human, it's just personal preference. He wants to die a human rather than be a mutant, and that's his choice. Much like the Tranquil in Dragon Age who commit suicide as a free mage rather than become a tranquil.

Q17: How will Virgil survive the Glowing Sea?

A17: He won't. He will likely move on to another area. He succeeds in making his reverse mutant drug, so he will probably go work on that somewhere else. Or maybe he hears the toot is destroyed and then gets to do whatever he wants.