Definitely to the first part. As for the second, it's not really a "special rate", it's just that you'll drop all coverages except comprehensive. You don't need liability coverages or collision coverage if your car is inside a covered building.

Now, if you happened to be parking it for six months on the street outside your parent's house, you would probably still want uninsured motorist coverage (in case someone ran into your parked car, but then just drove away).

Homeowners insurance does NOT cover cars inside a garage.