What is the relation between NP/NP-hard problems and Recursive/R.E languages? any of them a subset of another?
Every problem in NP can be decided in exponential time (and so it is recursive). Recall that one of the definitions of NP goes via witnesses:


  A language $L$ is in NP if there is a polynomial $P$ and a predicate $f$ computable in polynomial time such that $x \in L$ if and only if there exists a "witness" $y$ of length at most $P(|x|)$ such that $f(x,y) = \mathrm{True}$.


(A predicate is a function which returns True or False.)

Let's suppose that $y$ is in binary.
Given an input $x$, in time $2^{P(|x|)}$ we can go over all potential witnesses $y$, and for each of them calculate $f(x,y)$. If $f(x,y) = \mathrm{True}$ for any of them, then $x$ is in $L$, and otherwise $x$ is not in $L$.