Can we apply the Master Theorem to the following recurrence?
So, we have that $f(n) = O(n^c \log^k(n)) = O(n^0 \log^2(n))$; and since $log_b(a) = log_2(1) = 0 = c$, we are in the 2nd case of the Master's theorem (work to divide the problems is comparable to the work on subproblems (wiki)); thus, applying the theorem, we get $T(n) = \Theta(log^3(n))$.

On the floor operation, I don't know what is the cleanest way of solving it; one way could be to prove it for $n = 2^m$, and then extend to all $\mathbf{N}$ by monotonicity.

We can also calculate $T(n)$ directly:

$$T(n) = T(n/2) + log^2(n) = \sum_{j=0}^{log(n)} log^2(2^j) = log^2(2) \sum_{j = 0}^{log(n)} j^2 = \theta(log^3(n))$$