Homothetic preferences and weak separability With 3 goods (x,y,z), linear Engel curves, where z is separable of x and y, and with all first derivatives are positives and second negatives.

Does the demand of z change with the price of x or y? So, is z independent of $p_{x}$ and $p_y$?

I think so, because I have made some examples where changing prices didn't change the optimal amount of z. However I cannot find a formal proof.
EDIT: Seems to that after your most recent edit the answer is still no. 
A relatively simple counterexample is:
$$
U(x,y,z) = \sqrt{x} + \sqrt{y} + \sqrt{z}.
$$
This is clearly seperable and homothetic. But $D_z$ is is not independent of $p_x$ and $p_y$. I think this is clear just by looking at the utility function, as it is not of the Cobb-Douglas type. If you have doubts you can calculate $D_z$ from these equations:
\begin{eqnarray*}
MRS_{xz}(x,y,z) &amp; = &amp; \sqrt{\frac{z}{x}} = \frac{p_x}{p_z} \\
\\
MRS_{yz}(x,y,z) &amp; = &amp; \sqrt{\frac{z}{y}} = \frac{p_y}{p_z} \\
\\
m &amp; = &amp; p_x \cdot x + p_y \cdot y + p_z \cdot z.
\end{eqnarray*}
Then
\begin{eqnarray*}
m &amp; = &amp; p_x \cdot x + p_y \cdot y + p_z \cdot z \\
\\
m &amp; = &amp; p_x \cdot z \cdot \left(\frac{p_z}{p_x}\right)^2 + p_y \cdot z \cdot \left(\frac{p_z}{p_y}\right)^2 + p_z \cdot z \\
\\
\frac{m}{p_z}\frac{1}{\frac{p_z}{p_x} + \frac{p_z}{p_y} + \frac{p_z}{p_z}} &amp; = &amp; z.
\end{eqnarray*}
As you can see both $p_x$ and $p_y$ appear.