1. Do you mind if Rorschach randomly ends up in your apartment/house/office/wherever you might have a fridge? Be warned that it doesn't matter if you have any protections on the building, he'll only be coming in through the fridge, nowhere else.

Go for it.

2. Do you mind if he eats your food? And even if your character does mind, since Rorschach doesn't care, do you as a player mind?

All he has are Cadbury eggs and Red Bull. Go for it.

3. Rorschach always wears a mask and works as a masked vigilante. If he sees your character doing something he doesn't approve of, he'll probably put them on his list to check up on, which means he'll want to come back to the apartment/office again and again. Would he be able to drop in on your character at any time?

Sure.

4. So, if Rorschach talks to your character, he might offend them. Do you, as a player, have a problem with this?

Absolutely not. Do your worst.

5. If you do, then can we talk about what might be best, whether this means Rorschach doesn't tag your posts, or something that could tone down the wording, but not the intent?

LET HIM DO WHAT HE DOES, that is what makes him awesome.