Can you be sued above and beyond what the liability on your daughter's car insurance is if you are the primary on the co-signed loan?

It depends on who's suing you, a private individual pursuant to a crash or the insurance company. First, yes you can be sued as a co-signer. If the insurance comapany is suing you, the can only sue for specific damages. If an individual is suing you, the can sue above the specified amount, as many try to do for pain and suffering, aggravation, loss of wages. There are very few causes of action, however in a crash and the person usually only gets the original amount.