vector times cross product 
We have vectors $x, y, z$ where $z = x \times y$.

What is $x \cdot z$?
From my intuition, the cross product is perpendicular to both vectors, so dot product should be 0?

The geometric reasoning you mentioned is correct, and it probably the best way to understand what's going on.
Another option is as follows: We know that  $x \cdot z = x \cdot (x \times y)$. In general, the value of the triple product $u \cdot (v \times w)$ is given by the determinant that has $u$, $v$, $w$ as its rows. But, in our case, since $x$ occurs twice in our triple product, the determinant will have two identical rows, so its value will be zero. In short
$$
x \cdot z = x \cdot (x \times y) =
\det \left[\begin{matrix} 
\leftarrow & x & \rightarrow \\
\leftarrow & x & \rightarrow \\
\leftarrow & y & \rightarrow \\
\end{matrix} \right] = 0
$$
