Can a (vanilla) Kalman Filter's Observation Matrix $H_k$ depend on the state vector $x_k$? A vanilla Kalman Filter allows for a time varying observation matrix $H_k$. Is it allowable for $H_k$ to be a function of the system state $x_k$ in a vanilla Kalman filter? 

First, am I correct that this is not allowed in the vanilla Kalman filter? If it is allowed, then I don't understand how measurements would then act to correct the parameters of $x$ that predict them via $H_k$. I'd appreciate a more explicit description of this than my own hunch though!

Second, am I correct that the Extended Kalman Filter &amp; the Unscented Kalman Filter do explicitly allow for this situation? If I am, do these filters allow observations to correct the terms of $x$ that are predicting them?
The non-linear system model that the unscented and extended Kalman filters work with (omitting control) is the following:
$$x_{k+1} = F(k, x_k, v_k) \\ y_k = H(k, x_k, n_k)$$
where $F$ is the state transition function and $H$ is the observation function.  The linear Kalman filter is just the special case where $F$ and $H$ have the form:
$$ F(k, x, v) = A_k x + v \\ H(k, x, n) = B_k x + n$$ for sequences of matrices $A_k$ and $B_k$.  In particular, the functions $F$ and $H$ are linear.

The following is a bit of a pedantic distinction (particularly in the non-linear case) but while the expression $F(k, x_k, v_k)$ clearly depends on $x_k$, the function $F$ itself does not.  In the non-linear case this is no limitation.  In the linear case, if the matrix $B_k$ depended $x_k$ then the function $H$ would no longer be linear.