Tree fell on our car. I have a few questions regarding insurance?

We rent a house and the dead tree in the front yard fell over on my husband's car this morning. Our landlord said we had to call our car insurance company first, and if the adjuster (sp?) deemed the tree dead, then they could battle it out with his insurance, either way, we shouldn't have to pay for anything. Here is my question. Instead of hiring and waiting for a tree service to come out, our neighbor came over with a saw and took care of cutting down the limbs, pulling the trunk off of the car and breaking it all down so we could have it hauled off. Should our car insurance, or my landlords home insurance pay my neighbor? They would pay an actual tree/landscaping service to cut it up and haul it off, so would there be a difference? He worked hard and I would like to have him reimbursed for his trouble. Also, my husband missed work today due to all of this mess, will he be reimbursed by the insurance company? Just questions because we have never dealt with this before. Mature answers would be appreciated, thank you!