Can insurance company move a car to a repair shop without car owner permission?

No, the car is your property not theirs. The insurance companies have made special deals with body shops that give the insurance companies deep discounts. If you ask the shop to show you the agreement between them and the insurance, they wont. And for good reason, it would make you, "the CONSUMER" very angry! They usually agree to a lower labor rate and the use of cheap aftermarket parts, or used parts. The aftermarket parts have NOT been crash tested, but hey it's not their family in that car! the only problem I have with used parts is the insurance prefers used suspension, and they do not microflux the parts or do any kind of test for cracks, or other malfunctions. The reason the part is used is because the vehicle it came off of has more than likely been totaled. Just remember you are the property owner and you have a contract with the insurance company for reimbursment of damages or replacement, which ever is less.