They don't move much in a human lifetime but do move quite a bit on timescales of about 1 billion years. Since universe is about 15 billion years old galaxies will have made about 15 'orbits' in a cluster. Yes there is strong evidence that galaxies can collide and that this can strongly affect their evolution. When galaxies collide the individual stars almost never hit each other. Like 2 gnats in a baseball stadium.
Galaxies are mostly empty space.
Gas clouds do collide - causes star formation in some cases. Can make computer models to study how galaxy collisions work though. Our own Galaxy is a cannibal. Also MW is interacting strongly with LMC SMC - irregular since deformed by tidal forces. Will probably be eaten by MW eventually.