I believe they are saying that since the damage is to the interior of the condo, then it is the unit owner's responsibility to get any damages covered — whether they need to pay out of pocket or use their own condo insurance. The HOA "walls out" policy generally means that it will cover damage to the exterior walls, roofs, and other property contained inside the exterior framing, but not anything inside the unit.
The owner of the trailer would need to add you to their policy if you will be pulling their trailer regularly. If you will not be pulling it regularly, you should be covered — but I recommend you check with your specific company for policy details to make sure. If you have any questions, don't hesitate to ask.
You will need to contact the Department of Motor Vehicles for the state of Washington for information on registration. If you are required to register your car in Washington, then you would likely need a Washington insurance policy with your Seattle address.
No, the driver/owner of the car would be held responsible. If you have uninsured motorist coverage, you can have the damage to your car covered and you can file a claim on your homeowner's insurance for the damage to your garage door.
Because car insurance is regulated at the state level, you have to have insurance specific for the state you are residing in the majority of the time. You can buy a vehicle in another state but it will need to be registered and insured for your state.
He should be covered to drive your car as long as you have permissive use coverage. This means that someone can drive your car that does not live with you, as long as they have permission and it is not a regular occurrence.