1) I'd like to keep it to canon regions, but being a Gym Leader or E4 is open to discussion for basically any of the existing regions. Fair warning though, taking on a title like that for your character means you're basically a massive shiny target for the villains of the story. There will likely be some tough interaction right out of the gate.
2) The other teams and "bad guys" at this point have all been defeated as they were in the canon storyline of games; that being said though there is nothing stopping someone from deciding to try and revive them. If someone wanted to play a Team Aqua fanatic who's trying to recruit people to the cause and get their hands on Kyogre, I say go for it.
3) Your character can be basically anything to begin with, as I want this to be a very open and collaborative writing experience. Coordinators, breeders, professors, collectors, Gym leaders, it's all on the table. As for starters, you can have any Pokémon that doesn't evolve or any first stage evolution. For Gym Leaders and E4, your team must be discussed as obviously they're going to be strong and kinda stacked.
There will be multiple events taking place in different regions that all tie together, and at a certain point there will be a major event that brings most everyone together, obviously depending on their goals and such. Some characters may not be interested in that particular scene and end up elsewhere but everyone should intermingle eventually despite being in such a big world with so much going on.