Does $NP$-hardness of $c$-approximation (for some $c>1$) imply $APX$-hardness?
The answer for (1) is "unlikely".

It is simple to show (reduce from $Partition$) there exists no $\alpha$-approximation for Bin Packing, for any $\alpha&lt;\frac{3}{2}$, unless $P=NP$.

That said, Crescenzi et al. have shown that unless the polynomial hierarchy collapses, Bin Packing is not APX-Hard.

As for (2), perhaps you could phrase it as "Does not admit $PTAS$ unless $P=NP$".