Answer 1

Yes.

In Trainer Tower in Pokémon FireRed and LeafGreen there are three Trainers which have Shiny Pokémon. In the Single Battle mode, the seventh Trainer has a Shiny Meowth, while in Double Battle mode, the first pair of Trainers have a Shiny Espeon. Finally, in the knockout mode, the fourth Trainer will send out a Shiny Seaking.

Source

Answer 2

I remember there being a trainer in leafgreen that had a shiny meowth which was in the battle tower

Answer 3

Yes, there are sources that say that an in-game trainer has a chance of having a shiny pokemon. I can't remember where I read it, but when I do, I'll edit my answer and post it.