We traded in a vehicle with a lien on it. The dealership paid off the loan but the bank sent the loan paperwork marked "paid in full" and the title with bank signature to us. As of now, is the car ours? It has been over a month and we have not received the registration paperwork for the car we traded for. Should we keep this paperwork until we receive ours?

No the car isn't yours. Most likely the title has been transferred so the one you have is no longer valid. As to the reg on the new car, contact the dealer or your local DMV.