**No**, preimage-resistance or/and collision-resistance do not imply the infeasibility of finding fixed points in hashes.

For example, define $H(x)=\begin{cases}0^{256}&\text{if }x=0^{256}\\\operatorname{SHA-256}(x)&\text{otherwise}\end{cases}$

This is just as preimage-resistant and collision-resistant as SHA-256 is, yet verifies $H(x)=x$ and $H(H(x))=x$ for $x=0^{256}$.

Exhibiting a preimage-resistant and collision-resistant $H$, and matching $(x,y,z)$ such that $H(y\|H(z\|x))=x$ is almost as easy, and is left as an exercise to the reader.