When a car is paid off does the co-signer own half the car? Can a co-signer sign a quit deed for a car title?

Since the car loan is in joint name, presumably title to the car is in joint name as well. If so, then you are going to need to put in place an arrangement with the lender to remove his name from the loan and to get him to agree ( in writing) that he will cooperate with his name being taken off of the title to the car as well.  Simply by getting his name off of the loan does not guarantee that he will agree to sign the paperwork to remove his name from the title so make sure that any discussion between the 2 of you on this subject is confirmed in a writing.