Mine probably asks about wood stoves because we live in western NY, where wood is plentiful and winters are cold. Every insurer I ever had asked about dogs, because they can lead to substantial claims. Some dogs are considered to be aggressive breeds, so they charge accordingly. My garage cat does not count.
In NY we can register and insure a vehicle that is titled to someone else.
As far as parking on or under the lift, I don't think the insurance would care. The incidence of something happening with my lift in my locked building are far less than driving down the road or parking in a parking lot, especially overnight at a hotel.