Is it possible to have a saturated ideal on a successor cardinal which does not extend the nonstationary ideal? (i.e. some nonstationary set is positive for this ideal)
Yes, it is. The reason is that an ideal $I$ on $P(\kappa)$ is saturated just in case the quotient Boolean algebra $P(\kappa)/I$ satisfies the $\kappa^+$chain condition, and this is a property that is preserved by permutations of the underlying set $\kappa$. But the property of extending the nonstationary ideal is not preserved by such permutations, since we can perform a permutation of $\kappa$ that takes a nonstationary set to a club set. Thus, there are isomorphic versions of any saturated ideal that remain saturated, but which do not extend the nonstationary ideal. 

