I would like to prove the following three statements;
The first one I already solved:
$P(x,y|z) = \frac{P(x,y,z)}{P(z)} = \frac{P(y|x,z)P(x|z)}{P(z)} = \frac{P(y|x,z)P(x|z)P(z)}{P(z)} = P(x|z)P(y|x,z)$
However, with the second and third I struggle. The third one seems very intuitive and logic for me, but the academic proof is always a hard thing.
Could someone point me in the right direction for b and c?
Given that you proved (a), you can prove (c) as follows:
\begin{align*} P(e_1,e_2\mid h)=P(e_2\mid h)\times P(e_1\mid e_2,h)=P(e_2\mid h)\times P(e_1\mid h)\end{align*}
where the first equality is (a) (with $e_2$ in place of $x$, $e_1$ in place of $y$ and $h$ in place of $z$) and the second is the given assumption in (c).
To start (b), you have by the law of total probability (or your intuition) that
$$P(x\mid z)=\sum_y P(x,y\mid z)$$
Can you take it from here (of course, using (a))?