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.
In truth, no. Car insurance companies determine their rates based on their previous years' filings (loss to revenue ratio) in addition to other factors.
Your best bet would be to request a copy of your driving record from the DMV and see what is listed. If you see something that is not supposed to be on there you may be able to dispute the ticket with the courts.
A non-driving felony conviction most likely won't affect your rate, but it could make you ineligible for a policy with certain companies. Since you are already aware of the incident, you will want to disclose this when you call a company for a quote so they can let you know up front if that will create an issue.
Coverage & Claims
Whether you work in a different state than where you reside or are leaving the state on vacation, your insurance policy will follow you. Each state requires differing amounts of coverage so your current policy will adjust if you happen to be driving in a state that requires more coverage than the state where your vehicle is insured.