Insurance should take care of your auto damage if the other party is responsible and has liability insurance, or if you have collision insurance.
Towing and storage are part of a property damage claim. Be careful of storage fees because they can add up quickly. The insurance company of the at fault driver is liable for towing and continued storage, until they make a reasonable offer for your car. After that they usually won’t pay, so the tow truck storage company will then look to you for payment.
You may be entitled to a rental vehicle from the date of the accident until the at fault drivers insurance company makes a fair offer to repair or replace your vehicle.