You have pretty much hit and ran a vehicle. If their insurance has contacted you already, I would notify your insurance pay your deductible, but your insurance rate will go up pretty high. Your lucky that that's all they did. Usually they would arrest you for something like that.
Basically, you have to pay for insurance. If you get into an accident, you are at least partly responsible for the damage that accident caused, whether or not you were at fault. Insurance helps to pay for that damage.
Usually your own insurance.
NO
the insurance of the person responsible for the accident
Property damage liability car insurance will cover the individuals car and property that you hit. It will not cover anything to do with your vehicle if the accident is your fault.
If the damage occurred during the accident in question, then it should.
Call your insurance company and get a claim started.
They or their insurance company needs to pay for damage and medical bills in the accident. If they don't have insurance, they can still be sued for the accident by the victims insurance company.
It is advisable to call your insurance company immediately in any accident. If you do not, they may think that you are trying to defraud them by hiding the accident.
You are responsible for the damage you cause in an accident, regardless if you are insured or not. Having insurance transfers your responsibility to pay for the damage from you to your insurance company. If there are injuries to the other party, then the other party's insurance should pay for their injuries, but you are still responsible for the property damage you have caused the other person.
Insurance is VERY specific when it comes to when coverage starts. If you had the accident BEFORE you got insurance any damage sustained before you purchased the policy damage would not be covered. If you had an accident after the purchase of the policy then any damages would be covered.
This means that if the accident was your fault, your insurance will pay(up to an amount that is on your policy) for the other property and persons involved in the accident. Liability insurance does NOT cover your vehicle damage.