Will flying cars fall under aviation regulations?
Flying cars will need to be regulated by both the FAA and the DOT as per current laws here in the US. For what its worth flying cars are really more like road capable airplanes. When the car is airborne it will be subject to the regulations set fourth by the FAA for the airspaces it passes through and maneuvers it makes. When the car is on the road it will be subject to vehicular laws that any other car would see. That being said there is room for change here and who knows what rules may be passed in the future. There is also some grey area around things like landing in a parking lot, folding up your wings and driving away as well as the reverse case. 

The rules that currently govern aviation are more than capable of handling flying cars, even if they start taking off from peoples driveways. The larger issue is licensing the pilots. A drivers license is pretty easy to come by here in the states while a PPL or sport pilots license takes quite a bit of time. Everyone that currently would want a flying car would need both a drivers license and a pilots license of some form. 

As far as I know the FAA makes the rules for any and all altitudes. While some airspaces (class G) may be unregulated they are still considered from the ground up. There is no minimum altitude at which FAA regulations don't apply and its far from a free for all in the skys the way it is on the roads in a manner of speaking. 

Flying cars could potentially cause an increase in air traffic however the advantage of a plane (over driving) is generally speed. Until someone can build a car that can do 80MPH consistently on the road and a formidable 130Kts (and thats the slow end of planes) through the air the use case is pretty small. 

One of the biggest preventers in the whole flying car situation is the engine its self. Car engines fit quite a different role than aircraft engines. The similarities end basically at the fact they are both piston engines. Some car engines have been put in planes (Mooney/Porsche PFM 3200) and in the end they are generally not great planes. You can find a full discussion on this here. You also need to keep in mind that anyone who will be fixing these vehicles will need to be an A&amp;P licensed mechanic.

There are a few companies with some cool animations of future products and even a few test rigs that have been built but to date the engineering is just not really there. Chances are it will be some time before flying cars are even on the road let alone creating major air traffic issues. For the 279,000 a Terrafugia will cost you, you could buy a Used Piper Arrow (in great shape) and a Porsche and still have 79,000 left over for a hanger to keep them in...