Is a single carbon atom stable?
Is it possible to have a (free-standing) single atom of carbon?


It's definitely possible, but such an atom would be highly unstable and likely would have a short lifetime. An example of this happening is if you blasted some carbon material with a laser. Here's a really cool paper where researchers blasted carbon nanotubes and graphene with lasers and observed single-atom chains (not an individual atom in this case, but the same principle applies) of carbon exhibiting some very interesting electronic properties:

http://www.pa.msu.edu/cmp/csc/eprint/DT086.pdf


  If not, what is the minimum size of a carbon particle (diamond or
  graphite network)?


Buckyballs (60 carbon atoms) come to mind.

Tangentially related....

A few years ago, a team of Australian scientists fabricated a single-atom transistor (the basic building block of integrated circuits). The atom was Phosphorus. Granted, the atom was not free-standing, but still very impressive. Such a device is extremely difficult to fabricate, and requires cryogenic temperatures, so it's doubtful that this would ever become a viable technology.

http://www.nature.com/nnano/journal/v7/n4/full/nnano.2012.21.html