Whats the difference between a running shoe and a cross-training shoe?

Can you run a 5k in cross-trainers? Can you bike in running shoes?

Are cross-training sneakers good for walking?

I’m taking a power-walking class to fulfill my gym requirement and I wanted to get proper sneakers. Are cross-trainers good for walking… because I don’t want to get a pair of sneakers that I can only use for one sport?