On FPTAS and many one parsimonious reductions
Regarding number 1: The reduction to 3-SAT in the Cook-Levin theorem is many-one and parsimonious. There's no FPTAS for 3-SAT unless $\mathcal{P=NP}$, yet any problem in $\mathcal{NP}$, including those with a FPTAS, reduce to it in the fashion you describe.
With respect to number 2: I don't know many problems with FPTAS's but I strongly suspect that, if you look through a few and the reductions that prove them to be $\mathcal{NP}$-complete, you'll quickly find a counterexample. That's because I don't see any connection between being able to count solutions and being able to approximate them.