I pre-ordered the game, so I'm getting it on tuesday, while one of my friends didn't, and he might not get it for a week or two.
If I had a Rank 2 character, could I party up with him in a Rank 1 lobby and keep using the Rank 2, or not?
Or vice versa, would I be able to bring a rank 1 into a rank 2 or 3 lobby?

You can use your rank 2 but you can only use rank 1 abilities.

IIRC, one can temporarily down-rank themselves so they can play with lower ranked people without having the benefits of having a higher rank. How exactly what skills will be lost and which will stay with you will be chosen, I frankly don't know.

I think you might be able to be a rank 1 and play with people above your rank. I read about this a while back so I don't know if they took such a feature out. Of course doing this wouldn't be such a good idea until you reach rank 2 because at least then you can purchase class based abilities