Is the thermonuclear fusion of hydrogen to form helium operating in the cores of low-mass stars on the main sequence?

Answer 1

I can't remember if it is the proton-proton chain or if it's the CNO cycle... i'm pretty sure it's the CNO cycle, but i'm not 100% sure.

Answer 2

Yes. The fundamental energy source of all stars (even very low-mass ones) is hydrogen fusion. If an object isn't massive enough to begin and sustain hydrogen fusion, it isn't a star.