If a neighbor’s trampoline smashed the back of my car during a storm, are they legally responsible for the damages?

Your neighbor is liable for the damage to your car.  The insurance company might be claiming that the storm was an act of God and therefore the insurance company is denying the claim.

You can sue your neighbor for negligence.  Negligence is the failure to exercise due care (that degree of care that a reasonable person would have exercised under the same or similar circumstances to prevent foreseeable harm).

In order to prove negligence, you will have to prove duty (of due care mentioned above), breach of duty (failure to exercise due care by not properly securing the trampoline), actual cause, proximate cause, and damages.

Actual cause means but for the trampoline, would your car have been damaged?  If the answer is no, which appears to be the case, actual cause has been established.  Proximate cause means are there any unforeseeable, intervening acts which would relieve your neighbor of liability?  If the answer is no, proximate cause has been established.  The issue here is whether the storm hurling the trampoline into your car was unforeseeable.  If it was unforeseeable (act of God), your neighbor may be relieved of liability.  You should argue that the storm was foreseeable and that your neighbor did not adequately secure the trampoline.  Therefore, proximate cause is established and your neighbor is NOT relieved of liability.

Damages means the amount of compensation you are seeking in your lawsuit for negligence.  Your damages would be the cost of repairs to your car.  Depending on the amount of damages you are claiming, you may be able to file your lawsuit in Small Claims Court.  Your damages should include the cost of repairs to your car including the cost of a rental car while your car is being repaired if you need a rental car, and court costs.  Court costs would include the court filing fee and process server fee.  You will need to mitigate (minimize) damages by selecting an auto body shop whose charges are comparable to other auto body shops in the area.  If you select the most expensive auto body shop you can find, you have failed to mitigate damages and your damages will be reduced accordingly.  The same applies to the rental car.  You will need to find a rental car with a reasonable rate.  If you were to select the most expensive rental car you could find, your damages would be reduced accordingly.