TCAS on one aircraft reads the Mode C of a non-TCAS equipped aircraft and will provide an RA to that aircraft. The other aircraft will buzz along not knowing anything has happened?
Correct.
TCAS II needs Mode S because of the datalink abilities in order to co-ordinate RA's
Both aircraft have to be S-mode equipped (and working) to achieve coordinated RA. TCAS II can still make non-coordinated RAs against C-mode equipped aircraft and they're not moch of an issue, anyway, as chances that both TCAS computers will give same directions RA are pretty small. Even if it happens, soon there will be RA reversal in one with possible increased RA in other ac.
TCAS will provide an RA against aircraft with Mode C regardless of their Mode A (e.g. TCAS RA's against A7000/A0000 etc)
I'm not sure I have understood your question. To achieve RA, encoded altitude has to be received, otherwise it would be TA only.