Read your policy. If the policy says that anyone driving your car with your permission is an insured, then the company that insures that car pays. Many policies exclude certain drivers, so it is not a one size fits all answer.
The insurance for the car will pay, not the driver. Sadly, you are not required to pay anything, but if you are a responsible person, then you will....especially if it was your fault.
After an accident, all parties involved with the accident should exchange insurance information. Typically, the insurance companies will talk to each other about repairs and cost.
Insurance follows the vehicle, not the driver.
I would call their insurance, if you did not know who they are or they didn't have insurance, I would call your company and they can get the ball rolling.
The insurers of the driver who was deemed to be at fault for the accident.
Yes they will, but you may have to pay a deductable if you do not have uninsured motorist coverage, and you you must have comp & coll coverage.
This depends on the insurance policy. Usually your car is covered, no matter who is driving it. However, if you are driving a car and the owner doesn't have insurance, then your insurance would pay if you got in an accident.
Just because they left, doesn't automatically make them at fault. The insurance company claims department will make the determination base on the facts of the accident itself, not about what happened afterward.
The actual driver. Unfortunately, if the driver is your kid, and you're adding your kid to your insurance policy, it could affect your rates.
The driver at fault is always responsible for damages incurred during an accident. The at fault insurance company is responsible for damages to your car.
It would depend on whose fault it was.
Generally the lender, whose profits are all from the interest.