Bounding the gap between quantum and deterministic query complexity
As far as I know, the general bounds you state are essentially the best known. Changing the model slightly, Midrijanis has shown the bound that $D(f) = O(Q_E(f))^3$, where $Q_E(f)$ is the exact quantum query complexity of $f$; there are also tighter bounds known in terms of one-sided error (see Section 6 of this paper).

In terms of more specific, but still general, classes of functions, there is a paper of Barnum and Saks which shows that all read-once functions on $n$ variables have quantum query complexity $\Omega(\sqrt{n})$.

Although this progress has been limited, there has been considerable progress in lower bounding the quantum query complexity of specific functions; see this review for details (or e.g. the more recent paper of Reichardt, which proves that the most general version of the ''adversary'' bound characterises quantum query complexity).