The only A, B, C codes related to aerodromes that I'm aware of are the ICAO codes in Annex 14 and, of course, none of them have any bearing on the sort of strips that you've seen in my photos. But, yes, the broad concept of the "C" code, as explained by kishna is pretty close.
There are recency requirements for pilots operating into places like these and they also need a certain amount of animal cunning to interpret strip conditions prior to landing. This includes a requirement for knowledge of the route, or routes, to and from these aerodromes.
There's no "airfield certificate" for these places because the rules here don't require aerodrome certification as a distinction is drawn between aircraft, based on seating capacity. These aerodromes are served by aircraft with seating for up to 19 passengers and this is the cut-off point for certification under our rules.
The regulator places the responsibilty on the operator to ensure that the aerodrome is safe for their operations. This requirement has been responsible for operators ceasing services to some of these places.