Well, if I have time to change, I change in a room before meeting with the team. Otherwise, I'm usually only there in astral form and then it's just an illusion thing that's pretty much habit now.

....Superman changes in telephone boxes?

I don't know why they would fight and as far as I've know, they haven't.

Though it'd probably be Batman.

Batman is scary. Spider-Man's tendency to quip would probably just make him mad and scarier.