Can the cosigner take possession of the car?

Answer 1

Not unless their name is also on the title.

If you're not on the title then you only agreed to pay the loan if the person buying it doesn't. This doesn't give you any right to the car.

However, you can sue the primary borrower in civil court if they defaulted on the loan and you made the payments. The judge may order the primary to sign the title over to you.