Is there a regular tree language in which the average height of a tree of size $n$ is neither $\Theta(n)$ nor $\Theta(\sqrt{n})$?
I believe that the answer is as you suggest that no other asymptotics than $\Theta(1)$, $\Theta(\sqrt{n})$ and $\Theta(n)$ are possible. A promising route to prove this could be to apply the techniques from the paper which derives the $\Theta(\sqrt{n})$ asymptotics to the run trees of the regular language. Notice that a tree is accepted if there exists a run tree so it should be possible to first derive (using loc.cit.) the average height of a randomly generated run tree and take it from there, i.e. show that projecting away the states does not change the average height.