Normally but not always, a liability insurance follows the driver and comprehensive coverage follows the car. You should talk to your insurance company to find out the reason your insurance refused to pay for the repair. My guess is since you only have liability coverage, your insurance does not cover when you are not the driver. Under the circumstances your friend's insurance should cover. If she does not any insurance then she would be responsible for the damage to the other driver. You can also be responsible under vicarious liability theory and if that happens your only recourse is to sue your friend to get your money back.
The answer to your second question is so obvious.
Your third question is not a legal question. What kind of friend will leave you in a dire situation and run?
