Any update on this? Did you get the car back and is the insurance company taking action?
Never ever let a tow truck take your car to any place but the dealer, Carstar, or your own bodyshop you've used in the past.
If you take it anywhere else, they will charge crazy storage fees until the insurance company decides what to do.
I have seen forklifts being used to move the car around in the yard, which does a lot of damage.
The insurance company will pay to move the car afterwards.
Never ever let your car go to a storage yard, or unknown shop.