Is there a Time Hierarchy theorem for PH? Is it true that there are problems in the polynomial hierarchy solvable in time $O(n^k)$ (by an alternating Turing machine in some level of the polynomial hierarchy) that are not solvable in $O(n^{k-1})$ in any level of the polynomial hierarchy?
In other words - does there exist a time hierarchy theorem for the polynomial hierarchy like there does for P and NP?
If there does - a reference would be great.

The difficulty I ran into is that the simulating machine, when simulating machines from all levels of the hierarchy, is not in any distinct level of the hierarchy. Which leads to a related question - what is the smallest class such a simulating machine belongs to? Is there any sense in defining a class with $O(n)$ alternations (or $O(\log n)$ / $O(\log \log n)$)?
Yes. For example, the usual proofs of the time hierarchy theorem (by directly simulating arbitrary machines) can be used to show that for every $c \geq 1$, $\Sigma_c TIME[n^k]$ is not a subset of $\Pi_c TIME[n^{k-1}]$. The reason for switching from $\Sigma$ to $\Pi$ is that, in this diagonalization argument, we have to do the "opposite" of the machine we're simulating, so we have to run in universal mode when the simulating machine is in existential mode, and vice-versa.

You can also get a result like this without switching from $\Sigma$ to $\Pi$: for every $c \geq 1$, $\Sigma_c TIME[n^k]$ is not a subset of $\Sigma_c TIME[n^{k-1}]$. This can be done using the proof of the time hierarchy due to Zak (reference: "A Turing machine time hierarchy", Theoretical Computer Science 26(3):327--333, 1983). For an explicit reference to this version of the time hierarchy theorem, see Dieter van Melkebeek's "A Survey of Lower Bounds for Satisfiability and Related Problems" (available on his home page).