I have added some context. The Hamiltonian is time-independent.

I guess so. I am unsure if I could just do that or if I have to be cautious if superoperators are involved. I guess if dissipation of the form $\left(2 a \rho a^{\dagger} - a^{\dagger} a \rho - \rho ^{\dagger} a \right)$ is present I also just transform $a$ and $a^{\dagger}$?

@CosmasZachos, can you maybe give me a reference how the Wigner current for the environment is added? I understood from your book the that the part of the system orginates from the Moyal Bracket, but I dont see how I can include the environment analogously.

Yes, but $\partial_x V' = m \omega_0^2 x$, correct? I don't see how one of the $\omega_0$ cancels.

Probably, I was wondering because the same "typo" appears in equation (10) as well.