Is selecting some of the sets such that their intersection is of size k NP-hard?
As you suggested, the problem is equivalent to selecting sets with union of given size $k$. It is not easier than selecting given numbers (say, $m$) of sets with union of given size $k$, since we could add different auxiliary elements $a_S^{(0)},\ldots,a_S^{(n)}$ to each set $S$ and ask for a union of size $m(n+1)+k$, where $n$ is the size of the original universe. Now it is not hard to show the later problem is NP-hard because of a reduction from X3C (exact cover by 3-sets).