Does Bill have permission to slip you vague facts about your previous self or world?: Absolutely
Is there anything specific you would like him to focus on?: Maybe the fact that Rei is the last Martian, and it's guardian goddess?
Can Bill mention any random details of your world to someone else? : Go for it.
Is there anything you absolutely do NOT want him to know: Yes, the identities of her Canonmates. Any of them. Especially now that they're in game.