What classes of mathematical programs can be solved exactly or approximately, in polynomial time?
I can answer this part:


  Would it then be correct if I say there is an FPTAS for SDP?  I have not seen that stated anywhere, so that's probably not right.  But why?


The statement is correct, but we do not often see it because a stronger statement holds and is more important than this weaker statement.

An FPTAS is a polynomial-time algorithm which, given a problem and an accuracy parameter 1k, outputs a (1+1/k)-approximate solution.

But for SDP, the ellipsoid method and the interior-point method provide polynomial-time algorithms which, given a problem and an accuracy parameter 1k, output a (1+2−k)-approximate solution.  Note that the approximation factor is much better than what is required for an FPTAS.