You should be able to get your payment back, providing that you check with your insurance company to see if there are any fee disclosures. I would contact them and see what they can do to rectify the situation.
I do not know any insurer that offers a temporary suspension of insurance coverage. You could lower all coverages to the state minimum to make it cheaper, but if it is going to be more than a few months you may want to consider suspending the tags and discontinuing insurance.
Most financing companies will require you to have comprehensive and collision coverage which means you would not be able to get storage insurance. Check with your finance company to see what options you have.
Most companies will allow you to get insurance again even if you have canceled a policy with them in the past. Keep in mind that having continuous insurance coverage gives you a better rate so the price will likely be higher when you restart your insurance policy, and certain companies won't insure you if you have a lapse in coverage.
A statement of no loss is used when you are going to have a car insurance policy reinstated from being previously canceled. It's a form stating that you did not have an accident or any damage to the vehicle between the time the policy was canceled until the reinstatement date.