Insurers don't determine your actual cash value (ACV) settlement based on what you owe, but rather on what the car is worth just prior to the accident. Let's say you owe $20,000 on your new car, but it's only worth about $16,000. If your car is totaled, you might get a settlement check of $16,000 but still owe an additional $4,000 on your loan or lease.
The answer to whether insurance follows the car or driver depends on many variables, most notably the kind of insurance coverage being referred to. There are coverages that follow the car and coverages that follow the driver. In general, auto insurance follows the car instead of the driver, but the specifics of a claim can differ since insurance laws and coverage vary depending on the policy, coverage and state being dealt with.