Are subsets of $PSPACE$ closed under common operations
$SPACE(n^k)$ is closed under Kleene star. Just enumerating all possible partition (which can be done in linear space, or at most quadratic or $nlogn$ space), then decide each partition using the $n^k$-space machine.

This class is not closed under poly-time reduction. Prove by padding technique. This part would be found here at CS.SE or CSTheory.SE