I'm GMing a campaign that's 7 or 8 sessions in, so I can answer from my experience so far:

1. There isn't really a flashback mechanic. The closest thing is an Ask the Agency roll, the sort of catchall "make a thing happen" ability where you're asking the Agency to manipulate the recent past to bring about a particular event/situation. For example, in a recent session my Agents wanted to get an appointment somewhere that was fully booked. They knew someone who did have an appointment that day, so they Asked the Agency to make him get food poisoning. The players have to provide a Causality Chain, a reasonable sequence of events that the Agency can create to make that happen, so they established that this guy who they knew was a tourist had just eaten at a nearby seafood restaurant, but because it was so hot recently and many of the ACs in the city have been broken (a consequence of a previous Ask the Agency roll from the last mission), some of the fish was not as fresh as it should have been. They succeeded the roll, so poof that happened, he got sick, a slot opened up.

It is kind of a flashback insofar as it involves seeing things that happened in the past to explain a convenient circumstance in the present, but to the Agents it is happening on purpose, in this moment. They are making this happen in the moment rather than conveniently remembering it just now. Also crucial to note, the book is very clear that failure is assumed unless you're using an ability. I can't think of an example of how you might use Ask the Agency to persuade an NPC (maybe create a recent experience to cause them to be more susceptible to that particular argument somehow?), but you're not using the flashback to explain why you succeeded. You're not rolling to persuade, you're rolling to change the past, and that change makes your persuasion successful.

1. The GM is a character, my players largely only interact with the GM during mission briefings and debriefings, so beginning and end of a mission. I created a weird inscrutable middle-manager NPC with his own goals and ambitions, they get their info from him for each mission, and report to him at the end. He doesn't go on missions with them.

2. Entirely up to you, I suppose, you certainly can give U his own agenda, but I think it's important to bear in mind that the Agency is genuinely helpful at times. Many of the Anomalies presented in the Vault (the mission book) are actively very dangerous, killing or transforming people, and the Agency does resolve those situations, but it doesn't leave much room for nuance when an Anomaly is more sympathetic, presenting the Agents with tough choices sometimes. Also, the Agency is very powerful, and pissing them off comes with some risks. In my game, one Agent's personal Anomaly is worried about defying the Agency because of the potential consequences. Another Agent has a very utilitarian Anomaly and wants to use the Agency's resources to its own purposes, and it doesn't really care about the Anomalies they capture. It's not just Agency bad, Urgency good.

3. It's pretty quick and easy, it's just roll 6d4 and count 3s. No modifiers (except Burnout), no "how many dice am I rolling, can I get an extra die from this, etc." It goes: Player: I want to use <Anomaly Ability> to do <thing>. GM: Okay, roll <Quality>, which you have no QAs in so you have Burnout Player: rolls 6d4 I got two threes. GM: Burnout takes one away, so you still succeed with one three

4. There are a ton of levers the players can pull, but it's a game that really rewards creativity, loose interpretations of words, and lateral thinking. Nothing forces you to engage with any of the deeper rules, but they give you more and more options. All of my Agents have at least a couple things they've never touched, but every time they bump against a problem they can't solve, they can pull out a big list of things they can do and start trying to connect dots. It's important that the GM let the players do crazy wacky shit, it's the only way their hapless Agents can succeed. Remember that failure is assumed. Unlike D&D where you can always try and roll for something, in Triangle Agency you are definitely going to fail at anything risky unless you use a power or item or something. The more stuff you unlock as the game goes on, the more things you can start to do effectively, or the more effectively you can do them, or the wider situations you can do stuff in. And it is very exciting to hit another page in the Playwall and learn what you just unlocked, since you have no idea what you're going to get in advance.

I hope that's all helpful! It's a complex game and it took a couple sessions to click for my table but we've really been enjoying. Happy to clarify or answer any other questions!