How can a car accident from almost three years ago keep adding costs?

You need to confer with a local lawyer. The cost of the meeting will be well worth the cost of the additional money the insurance is demanding.

Who is responsible for an at fault accident in New Hampshire if they are uninsured? The Owner of the vehicle or the driver.?

They are both potentially liable for the damages.

It depends on the terms of your policy as to when coverage began, immediately or upon payment...?

Not sure why they need a W-9, why they are paying less than the estimate or what a non-vehicle car incident means.

Can a finance company do this?

The answer is within their contract. I do not foresee a reasonable resolution without counsel.

False claim (against me) listed on my car insurance.

It is well beyond the 3 year look back period to determine rates. This is much ado about nothing.

Talk to local counsel. However, it sounds like your insurance is just trying to recoup some of their money from the other driver.

Can I do something about an insurance company going against a contract?

Send the check and sop worrying about it. It is not worth the hassle of arguing over it.

You are going to need to get your license reinstated with a SR 22 bond likely. Contact the DMV regarding the process.

If a car insurance claim was opened by other party and it was a hit and run why is suspect not apprehended yet by law enforcemen

No idea and there is no way to say what the police know. Unless there was a fatality or serious injuries, the police probably do not have the time...