Can Katherine compel your character? Can she attempt it? Yes. Will it work? Maybe. River's brain is a literal mess with pieces missing and there's still futuristic brain washing results battering around up there. As with everyone else whose characters have a mind control ability, I'll say that Katherine is more than welcome to try but the results will vary wildly each time.
Can she bite your character? She can. She shouldn't. River would try to kill her and since it's unclear whether or not she's been exposed to PAX in canon, I'd rather not spread her blood around in game.
Could her blood heal your character? Again maybe but not certain!
Is your character able to injure her? It's possible! It would probably depend in what weapons are at hand.
Is Katherine able to injure your character? Yes.
Could Katherine come and meddle with your dreams? Yes! Though she tends to have other people's dreams too.
Would she notice anything strange about your character? She's fairly human so nah.