everywhere as long as your on your own land or you have the permission of the person who owns it
Hell no, 50cc scooters can't even ride on there! If it doesn't have a motor or transmission, it would be unwise.
No, its illegal to ride a dirt bike on the road or sidewalk in all states.
There's no solid answer to that. Some places do have mandatory helmet laws, others don't care. Some paths CAN be pedestrians only, while other paths are for dual use. You have to check your local...
Check with the DMV because they can tell you what vehicles are legal to drive in the state and what state laws apply to them.