There is no straight-forward answer to what building materials you use for your house and the homeowners insurance rate you will end up with. Here's a tip to keep in mind that may help you make a decision: brick, stucco, and concrete are preferred by most insurers because they tend to be flame-retardant so the risk of losing the whole building decreases.
Yes, you can file a claim with your homeowners insurance but I would see if your lawnmower's warranty is still valid before you look into a claim. If you do decide to file a claim, it would be covered under personal property.
Unfortunately, if you start the claims process, the insurance company can list it on your CLUE report through LexisNexis which contains your claims history — even if there was no payout or resolution. You can dispute with State Farm to see if they will remove it, but at this point, that is really the only option which is definitely frustrating.
You should have no trouble getting mobile home insurance to cover your home including the add-ons, and there are several companies that will insure that type of home. You will need an HO-7 homeowners policy that specifically covers mobile homes.
Insurance companies (depending on your policy and whether you have replacement cost value coverage) are obligated to bring your property back to its previous state, but no better. This, in theory, means that it should be no worse as well.
I'm sorry, but this is a policy-specific question since the vacancy clause is an endorsement (addition) to your policy to add further coverage. You'll need to contact your insurance company for a specific answer.