Randomized and deterministic reduction
Let's answer your questions one by one:


A deterministic reduction proves NP-hardness. A randomized reduction doesn't. If a problem is NP-hard with respect to randomized reductions, then it could (potentially) be solvable in polynomial time even if P$\neq$NP. The assumption BPP$\neq$NP does rule out polynomial time algorithms for any such problem, however – even randomized ones.
The consequence is that P$\neq$BPP, which is conjectured to be false.
NP-hardness is defined with respect to deterministic reductions.
We say that the problem is "NP-hard with respect to randomized reductions".