If it damaged your house or another covered building, policies typically pay a limited amount to haul it away, commonly $500 to $1,000 per loss. If it hit nothing, often little or nothing. Many (not all) policies make an exception when it blocks a driveway or disability ramp.
Usually you file with your own insurance company. It may go after your neighbor to get its money back if they were careless.
Generally no. That's usually covered by the comprehensive part of your car insurance.
It can make things harder. The owner may be blamed for the damage, and insurance companies may look closely at whether the tree was cared for.
If wind brought it down, it may. In Alabama that deductible may be a percentage of your home's coverage amount or a flat dollar amount.