If a rented car is stolen and insurance pays but then the car is recovered what happens to the car?

The finder is obligated under civil and criminal law to notify the Police and the Owner of the vehicles recovery and location so they can come pick it up. Since the the Insurance Company already paid for it. It belongs to the Insurance Company. If the Rental company wants it back, They will have to try and buy it from the Insurance Company.

All salvage rights would belong to the Insurance Company because they have already bought and paid for the car.

It would now belong to the Insurance Company. If the Rental Car company tried to keep the vehicle without first negotiating a purchase or buy back from the Insurance company it would be considered Grand Theft. They would be subject to criminal and civil charges.