Must I compensate my brother-in-law for a car I borrowed from him that was stolen and then returned damaged, after he refused to repair it and sold it for a pittance, then came back demanding its value from me?

The loaned item is a trust in the hand of the borrower, who is not liable for it unless he exceeds the permissible use (ta'addi) or is negligent (tafrit), or if a guarantee was stipulated upon him. So, if you were not negligent, nothing is incumbent upon you. However, if you were negligent, then you are liable. If the car is returned, you are obligated to repair it so that it returns to the condition it was in when it was borrowed. If the car was sold, you are obligated to cover the difference between its value at the time of borrowing and its value after its return. There is another opinion that mandates a guarantee absolutely, even if there was no negligence. It is advisable to refer to scholars of knowledge and religion to resolve disputes in a way that satisfies both parties.