Backdating: Things with me can drag on for a long time due to timezone issues, so this will be invoked a lot.
Threadhopping/jacking: As long as the thread isn't locked or filtered, yes.
Breaking the 4th wall: Dropping hints about his position against the mafia or anything like that is okay, but nothing about him being a fictional character, please.
Offensive subjects: Pineapples. Comparing him with the mafia, mostly.

Hugging: Go ahead, but he probably wont reciprocate unless he's pretending to be a nice guy.
Flirting: Go ahead! He'll flirt back too, with more innuendo and such, but more often than not, his flirting wont have any real interest behind it.
Kissing: Sure, but he will have problems kissing back.
Sexual situations: Totally okay. Mukuro prefers 18+ people, but OOCly my hard limit is 15, so I can roll with younger people too.
Romance/relationships: Why would you even-- You can try, but he's hardly interested in those kinds of relationships. If he enters a relationship with someone, it's probably because they're queens or at least bishops instead of pawns.

Mind reading/control: Your character can certainly try, but since that's his shtick, he's resistant, and not going to be happy when he finds out.

Violence: Yes please. We can be as violent as possible, since Mukuro has no qualms about kicking someone's ass through the six hells if he has to.
Injury: Sure, get him as bloody as you like! However, removing a piece of him is a no.
Murder: No.

Anything else: Possession-related permissions are here.