Do I have the right to choose my repair shop even if my insurance company is paying for the repair?

Yes. You have the right to have your car repaired at any shop you choose. Your insurance company is required to pay the reasonable cost to restore your vehicle to pre-accident condition according to the coverage you purchased. Even if your vehicle needs to be towed, you can still choose your repair shop. The insurance company is responsible for paying for your repairs and should be responsible for paying the tow bill to the shop of your choice. If the insurance company selected a shop, and you prefer to go elsewhere, they should be obligated to pay for the additional tow.