Does NP-hardness imply P-hardness?
No such implication is known.  In particular it may be that $L \ne P = NP$ in which case all problems (including trivial ones) are NP-hard under poly-time reductions (as the reduction can just solve the problem), but trivial ones (in particular ones that lie in L) are surely not P-hard under logspace reductions (as otherwise L=P).  

The same goes for NC instead of L.