None. A cosigner is entering into a legally binding contract to repay the debt if the primary borrower defaults on the lending agreement. The cosigner does not have any other obligation nor ownership...
By co-signing the loan, they are guaranteeing that you will repay the loan. They do not need to be on the auto insurance policy, but it would be in their best interest.
Yes, because you have to list all owners of the vehicle on the policy. Technically, an owner of a car can be sued and held legally responsible for an accident or occurrence, even if they're not...