Are there class C amps?Yes, but they are not used in audio:
http://en.wikipedia.org/wiki/Electronic_amplifier#Class_C
Also, re "Digital Amps vs Solid State Amps," I'm not sure it's clear that "solid state" has nothing to do with amplifier class. It means that the amp uses transistors, as opposed to tubes. I don't think there are any Class D amps that are NOT solid state.
Also, a true "digital amp," such as the TacT's, has a digital input and a signal path that is entirely digital until d/a conversion is performed, at a point in the signal path that is very close to the amp's output. See http://www.tactlab.com/Products/MS2150XDM/FutureofAudio.html
Regards,
-- Al