Number of levels in the recursion tree
If $b$ is not a constant, then no, you can't say the number of levels is still $\log_b n$.

In your example, we have $T(n) = a T(\sqrt{n}) + c$ (since $n/\sqrt{n} = \sqrt{n}$).  There are more than 2 levels in the recursion tree.  In fact, there are $\lg \lg n$ levels in the recursion tree.  So, no, you can't just compute $\lg_{\sqrt{n}} n = 2$ and conclude that there are 2 levels in the recursion tree -- that gives the wrong answer.

(Why $\lg n$?  Suppose $n=2^k$.  Then each level halves $k$, so we do a total of $\lg k$ levels we get down to a constant.  Since $k = \lg n$, we get a total of $\lg k = \lg \lg n$ levels.)

The Master theorem gives a formula for solving this kind of recurrence, but it's only valid when $b$ is a constant.