Most insurance companies will pay when a tree falls on your property. By property I mean a fence, building, shed, piece of equipment or something other than just falling to the ground. That is unless it falls on the ground and blocks your driveway, then they will remove it too.
They will try and get out of it by saying trees and bushes aren't covered but most policies have a clause in them that will not only remove the tree from what ever it fell on but to also remove the whole mess from the property stump in all. Read your policy to be sure before you call so you will be ready. Most people around her normally have $1000.00 deductibles on their policies but that is the total damage of the claim and not for each item so you have enough damage to get them to fix this. You just have to figure out a way to get your deductible covered. Tree companies around here put $2500.00 to $3000.00 a tree, for trees around the size of the ones you now have on your man cave and that is for cutting it off the building and then removing it from the property. The stumps are then ground down to nothing but dirt and chips.
If you feel comfortable working with a chain saw and are not afraid to burn the brush and wood, you can use that to burn up the stumps yourself and save your deductible. You can probably work out a deal with them to give and estimate for the total cleanup then work out a deal with them to just cut it off the building and leave the rest for you to take care of.
I had the same problem a year or two ago only about 10 times worst. I had almost 90 huge oak trees blow over with 29 of them landing on different things from the roof of the house, my partially constructed pole barn, well, fish pond, tools and equipment, trailers, driveway. They paid to have every one of them removed accept the ones that just fell in the yard and not hitting anything. I paid a tree man to give me an estimate and then I did the work myself. I cut up the wood in long lengths, that is the stuff that I didn't burn up burning the huge stumps and when It was over I had enough fire wood for ten years or more and also enough money to do some improvements around the house and some additional additions to my pole barn.
Don't forget they will replace your building and it's contents or anything else that was damaged, so make sure you list everything that was in the building and don't slight yourself no matter how worthless you may think some of the things are, put them down and put a number on it because you are probably not going to get full price for anything until it is replaced. Use the junky stuff as throw aways so when you go shopping you will get enough to at least replace the good stuff and maybe even up grade a few choice items.
The main thing is don't slight yourself and under estimate the damage because they are going to do that for you with depreciation so don't help them slight you. Dealing with insurance companies is scary business to some people and they will do what ever the can to cut their cost so it's up to you to get what you have coming and not worry about their bottom line. your bottom line is what you need to worry about. I'm not saying to lie or try an cheat them just get what you have coming within the limits of your policy. And, don't worry about them going up on your insurance or cancelling your policy, they wont do that with a wind damage claim because it's obviously through no fault of your own. Oh.. they will however probably go up on your insurance a little but they will do that no matter what the claim is, little or big so you may as well get your monies worth. Read your policy and know your rights and get what you have coming to you not just what they think you should get.