Re: FAA enforedes airspace restrictions to ground level
They do. Their authority attaches to the aircraft, not the air. As long as it flies and is larger than, say, a foot in dimension, they FAA holds legal authority via various acts that define its authority. That's why FAA regulations apply to aircraft even when they're on the ground.
Where authority over the air comes in is that the FAA can regulate what can go into regulated airspace. So, for example, aircraft that can climb over 500 feet need to follow FAA rules regarding flight plans and so on, so as to reduce the risk of incursions and collisions. They don't have such controlling authority at lower altitudes, but they still have a say over the aircraft themselves.