The experience you have is going to depend on the production company you work with. I'm a car guy, so when we rented gorgeous a 1920 Model T stake bed truck for a shoot, I personally chose the flatbed driver and made sure that everyone on the crew understood the importance of treating the car as carefully as Nitro-glycerine.
If you want to allow a production company to use your car, the precautions should include photos of every surface of the car and close-ups of the paint before you start. That protects both of us. Of course comprehensive insurance. And, the language of the contract should make it clear who gets to decide if any repair is acceptable. The most important thing I can say is that at the end of the day, a car is only tin and rubber. If you can't get your mind around that, don't rent the car out.