Yes. Actually, you could bring an action against the trimmers for the entire damage, but then you would have to turn over all but $1000 to the insurance company.
Yes. Before you withhold anything, contact the tree trimming company, explain what you are doing, and try and get them to agree. If they won't agree, then send them a letter explaining the deduction with a check for the balance. That might not stop them from suing you, but will help set up a good defense.