Insurance is regulated on the state level and every state has its own insurance laws. For example, Michigan residents are well-known to pay the most expensive premiums in the country due to the state's coverage requirements, which include unlimited medical coverage for the lifetime of car accident victims. Interestingly, our data suggest that the average rate in Washington is slightly less than the average in Texas ($1,368 versus $1,415, respectively).
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.