Who would pay for damage to your home that was caused by a friend's car catching fire in your driveway?

Turn it in to your homeowners insurance and have them work out the details. Most likely they'll go after the friends insurance, but that's their job. Let them figure it out. It's your house, so your insurance should pay for damage caused by an unforeseeable accident. If someone caused the fire, or permitted it to spread to your house through negligence or reckless conduct, then the insurance company can attempt to sue that person for reimbursement.