$L^{\prime}=\{x \# y \mid x y \in L, y x \notin L\}$ where $L$ is regular
You can write this language as the intersection of two languages:
$$
\{ x \# y \mid  xy \in L \} \cap \{ y \# x \mid xy \notin L \}
$$
I assume you know how to handle the first one. As for the second one, construct an NFA that acts as follows:

Start with a DFA for $L$.
Guess which state the DFA will be on after reading $x$.
Based on this guess, verify that after reading $xy$, the DFA reaches a rejecting state.
Now read $x$ and verify your guess.

You take it from here.