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.
Once you know the approximate value of your car and the cost to carry collision coverage, then you can make an informed decision about purchasing that coverage. Many people find that it's a good idea to cover newer cars, but as cars get older, their values decrease, and you might consider omitting or dropping this coverage to save money on your auto insurance.
Comprehensive car insurance covers damages from an "act of God," or events that are not caused by a car driving into something else. An "act of God" can include things like damage from a heavy tree branch falling on your car. Since you have no control over when or why a tree branch would fall on your car, this kind of accident would be covered under your comprehensive policy.