expected number of balls in k emptiest bins

This is only a very partial answer, and I would have considered putting it as a comment if I were a more reputable contributor, but I hope it's at least somewhat helpful.

Consider the simplest possible non-trivial case, $p=2$ and $k=1$ -- i.e., you're looking for the expected number of balls in the "lighter" of two bins. It's easy to see this is related to asking for the expected distance to the origin of a random walk, which, if I remember correctly, is asymptotic to $\sqrt{2n/\pi}$. So the expected number of balls in the smaller bin is asymptotic to $n/2 - \sqrt{n/2\pi}$.