Here's what I know, someone correct me if I'm wrong:
CATIII refers only to ILS (Instrument Landing System). There are CATI, CATII, and CATIII ILSs, and the higher the number, the less stringent the weather requirements are (and the more stringent the equipment requirements are). CATIIIb allows landings with no visibility at all.
Airports are not CATIII certified, specific approaches are. In order to fly a CATII or III approach, the airplane and flight crew must also be certified. Most major airports are going to have either a CATII or CATIII ILS approach. For example, JFK
has a CATIII approach to 4R and 22L (both ends of the same runway). The ILS to 13L is CATII, and the ILSs to 4L, 22R, 31L and 31R are all CATI. The smaller airports will not have a CATIII approach, as they only become useful when the visiblity is REALLY bad, and the expense isn't justified.
I don't know of any list of airports with CATIII approaches.
RNAV is a method of navigating while enroute, and can also be used for approaches (though with far less precision than even a CATI ILS, and thus RNAV approaches are more stringent with regards to weather requirements). There are several types of RNAV, including VOR Radial/DME
(not sure of the technical term for this, but it basically takes two VORs, calculates where you are in relation to the two of them by measuring your bearing and distance from both, and uses that to plot a straight course to a position you set in, which will be defined as another bearing and distance from both of them), inertial navigation systems, or GPS. Basically, RNAV is anything that allows you to navigate accurately on any course of your choosing, as opposed to being restricted to tracking navaids.