Even then because they are considered living creatures obtained through a special event. (Essentially, if no one claimed them they would just starve in a ball and we don't want to deal with that, particularly since there have already been a number of dropped characters' Pokemon. If their owner goes home, they go home to the PokeWorld.)

Yes.

Yes, that's fine as long as the character-owner is still in the game! Just be aware that high-level Pokemon will be less likely to listen to anyone who isn't their master.