Bounding the gap between quantum and deterministic query complexity Although exponential separations between bounded-error quantum query complexity ($Q(f)$) and deterministic query complexity ($D(f)$) or bounded-error randomized query complexity ($R(f)$) are known, they only apply to certain partial functions. If the partial functions have some special structures then they are also polynomially related with $D(f) = O(Q(f)^9))$. However, I am mostly concerned about total functions.

In a classic paper it was shown that $D(f)$ is bounded by $O(Q(f)^6)$ for total functions, $O(Q(f)^4)$ for monotone total functions, and $O(Q(f)^2)$ for symmetric total functions. However, no greater than quadratic separations are known for these sort of functions (this separation is achieved by $OR$ for example). As far as I understand, most people conjecture that for total functions we have $D(f) = O(Q(f)^2)$. Under what conditions has this conjecture been proven (apart from symmetric functions)? What is the best current bounds on decision-tree complexity in terms of quantum query complexity for total functions?
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).