Yes, you are still responsible for whatever amount remains unpaid on your loan.
You would check out the written policy to see which it was. Accidental insurance is much cheaper than the other, because dying by accident is less common.
There are several different options in order to get an Accident Cover insurance. Depending on the country one lives in one might check for example "Aviva plc" (British insurance company) or "eHealthInsurance" (American company) - just to list two of them.
It depends on if you have a lien placed on your vehicle (by you lender. If you own the car free and clear, you can take the insurance money and not repair the car, but if there is a lien on your vehicle more often than not the insurance company will make the check out to you and your bank.
If the other company only gave you the deductible amount, then probably not. But if both paid you the full amount, then you should turn the other company's check to yours. If the larger check came from the other driver's insurance, I'd return the check from your own insurance company - you're entitled to the deductible amount, if you're not at fault.
You need to check with your agent or policy services for you company. They can tell you their rules for accident/claims related rate increases.
Always if in California. Check your local DMV office. But many people do not and the insurance company will figure it out anyway.
That depends on the insurance company. Some will send you a check, and others will only pay the shop.
Need to check the state law for the statute of limitations for property damage claims.
Not all car insurance plans provide loaner cars when your car is in the shop after an accident. This is a convenience which may or may not be offered. Check with your insurance company to see if this service is provided.
Each insurance company has their own policy, therefore you would need to check with that particular company. Was there an accident? What state did the violation occur? Do you have a license for that state?
They cut you a check and keep your car.
You might get better insurance rates if they don't find out you're lying. A new insurance company will likely check your driving record with your previous insurance companies though.