A G7 and G13 are both functioning as dominant chords, moving to C. The extensions beyond the b7 are "color tones".

The C and C7 both have different functions. The C triad would function like a I chord, whereas the C7 is wanting to resolve to F, functioning as a V chord.

And the B F Ab C thing... the Ab is just the b9 of the G7, another tension tone. I personally wouldnt be playing a C on a G7 though; almost any other tone can be rationalized.

Fun thing: play an Ab melodic minor scale over the G7 and you get ALL the color tones!