Stationary sets and $\kappa$-complete normal ultrafilters

Asaf's answer is totally right, but let me also point out that you don't even have to go to a special model to see that your conjecture fails. The point is that sets in any normal measure on $\kappa$ must reflect many properties of $\kappa$ itself (since these sets $X$ are exactly the ones for which $\kappa\in j(X)$, where $j$ is the ultrapower map). For example, the set of regular (or inaccessible or Mahlo etc.) cardinals below $\kappa$ will be in any normal measure on $\kappa$. This means that the set $S$ of singular cardinals below $\kappa$ (which is stationary) is omitted from all normal measures on $\kappa$.

No. Work in $L[U]$, the canonical inner model, then $U$ is the unique normal measure on $\kappa$. Pick any $S$ such that $S$ and $\kappa\setminus S$ are stationary, and then only one of them can be in a normal ultrafilter.