| Your auto ins needs to pay
I worked as a Property/Casualty customer service rep and sales agent for American Express previously. I was licensed in insurance in over 46 states.
Regarding the garage door, this is an expense that your auto insurance is liable to pay. You should not pay this out of pocket, as this is a covered benefit under your liability portion. One thing to consider, however, is the cost of the garage door. Any time your insurance pays out on a liability claim, you can guarantee a rate increase the following year or renewal term (The mere fact that there was an accident will also cause this, however the amount they are allowed to rate you up or whether they are legally allowed to rate you will depend on the amount paid for damages.) The property damage sustained by your vehicle is covered under your collision coverage, while the garage door is covered under your liability. You could file with your home insurance, but be sure to advise them that this was a loss due to a motor vehicle accident and provide all your auto insurance info. They will then subrogate for you from your auto carrier.
If your auto insurance refuses to cover this, you need to contact the insurance commissioner. This is illegal and poor business practice. | 
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