I love kitties. Though it's nothing heavily serious for us, my little one on occasion will just start acting like a cat (rubbing his face on my leg and purring, curling into a ball and wanting to be pet). My role in that is to simply pet him if I feel like it or ignore him if I don't, though he does like to be kicked out of the way if I need to walk where he is. Also for us it's just a game; we don't do that during sessions so I can't help you there.

Punishment for cats is frequently being sprayed with a bottle of water (cats really do loathe being wet), rewards are licking milk from a dish or being fed bits of meat from the table.

Though more difficult than training dogs, it is possible to train a cat (it just will always have an attitude and may or may not obey you at any given time).

Cats are far more independent than dogs in their behavior in that they basically do whatever they want- so fortunately for you, there isn't all too much for you to do (have you ever owned a real cat? If you haven't, you might consider getting one just to learn what they are like, though that's a big commitment, too). However a cat cannot feed itself (if it's indoors only), so she would have to depend on you for that if you wish; with online play I guess you would determine when she may eat, which would probably be whenever you get home from your day, however late that is.

I'm sorry, but I don't know how you'd do pet play online at all; animals can't talk, so they'd have nothing to type. Well, you could task her, I suppose, but she'd have to "turn human" to report on it. Good luck with it, anyway, it sounds like you're having fun!