Does companion equipment deteriorate, do I need to repair it? Does giving them different kinds of ammunition do anything? I currently have Cass and she's always using her shotgun, except when I order her to go melee. How can I make her using a different gun?

Companion equipment does not deteriorate, and they don't benefit from custom ammunition types. To make Cass use a different weapon you'll have to find one that she can use to output more DPS than she can with her default shotgun, and you'll have to keep her supplied with ammo for it.