As the title says, I am curious does it support crossplay between those two platforms? Is multiplayer only about battles, or does it support campaign too?

I was able to play with my friend who owns the game on Steam with no problems. You just may not be able to see their game on the game browser but they will just need to send you a game code for you to join.

Also the multiplayer is on the full campaign map, and from what I experienced you can not play battles in multiplayer campaign.