Is there a minimum distance between a runway and a public road? Does the FAA require a minimum distance the between the TDZ of a runway and a public road? 

The reason I ask is because an aircraft landing at a small GA airport (Garnett Municipal, K68) came across the roadway I was driving on low enough that if I'd been 200 yards up the road he'd have hit my truck. Or at least he'd have had to go around. 



From the satellite pic you can see that the end of the runway is about 30 feet from the road. The TDZ is about 150 feet. A 3° approach would put you just short of 8' AGL at the road, which is just about exactly how high he was. My truck has a clearance of 10', so it's quite possible to strike vehicles on the road if one is too focused on the runway.

Although it's not a highway (that's about 1000' from the runway), it's the only paved road going east/west for several miles so it gets a bit of traffic

Is it common to put a runway so close to a road?
Yes, there is a minimum area that must be free of obstructions and closed to vehicles, but only for airports that have air carrier operations (see 14 CFR 139.309) or that receive FAA grants. AC 150/5300-13A - Airport Design has all the technical details; the actual distances are calculated from formulas, so there's no single answer. If you're really curious, you can read all the gory details in Chapter 3 and this FAA FAQ has a (very) simplified explanation:


  What is the size of the RSA [Runway Safety Area]?   
  
  The RSA is a rectangular box surrounding
  the runway and is based on the runway design code. The dimensions
  range from 120 feet to 500 feet in width and 240 feet to 1000 feet in
  length beyond the departure end of the runway. (see FAA Advisory
  Circular 150/5300-13, Airport Design) Generally, on airports that
  serve air carrier aircraft, the RSA extends 250 feet either side of
  the runway centerline and 1000 feet beyond each end of the runway.


You didn't mention which airport you're asking about, but presumably it's either privately owned and/or doesn't receive FAA grant money.