My dad is an insurance underwriter.
THEY have to file with THEIR insurance for the shed regarding your tree. You are not responsible for damages to their property that your tree caused because it was due to weather. This is a FACT.
You need to file a claim with your insurance company for your car for damages and report her car as the one that caused it. Don't make the mistake of not reporting it. You'll regret it, I promise. And under no circumstances should you let someone that's their friend work on your car.
I live in CT and we got hit pretty hard by the storm where I live. My cherry tree landed in my neighbor's yard and nearly hit one of their cars. Thank goodness it didn't, but we would NOT have been liable for it. We did help clean up though just because it's neighborly and we like these neighbors. But I also took photos of the damage so that they can't claim something later on that didn't happen. My father didn't raise a stupid daughter.
EDIT: Please do NOT give them $100. You are not liable for damage to their shed. They are. They HAVE TO FILE with their insurance company because it's THEIR PROPERTY. Just because it's your tree, that doesn't make you liable. The damage was caused by weather, not by you. Even if their insurance company won't cover the cost to repair or replace the shed, you are not to give them a red penny.
I know what I'm talking about here.
And you MUST report the car incident especially if you know for a fact that it was them whether or not the police were involved in a report and in spite of whatever "agreement" you made with them prior. These neighbors are only looking out for themselves and, in my opinion, not being neighborly or friendly to you at all. You are not obligated to do what's best for them, which is all they're trying to do. Go through insurance the way you're supposed to, and tell them to go through theirs for the shed. Don't hand over a penny to them. You'll regret it.