I have a question, with Rebirth can I make a deck with The Professor: Keeper of Knowledge. Using his ability then swap him with say Kate "Mac" McCaffrey: Digital Tinker and use her. Does it matter if the deck is too small for their min deck size or the influence exceeds, the new ID's. With now owning Rebirth and seeing useful for situations like this I just want to know if this is viable or is illegal to do in the world of Netrunner.

Your deck has to be legal before the actual game starts. So yes, you can rebirth with the professor into any other shaper identity, withou caring about influence or deck size. Kate is always a good rebirth target, as well as Kit, and maybe even Jesminder Sareen in a NBN-matchup. —
Do you think that rebirth is legal only in 46 card decks? ^^ Nope, it's fine to have illegal (lower then decksize or extrainfluence) deck during the game. Deck check is done only before the begining. —