For long range interactions, in general you are better off using TDVP, since TEBD requires using swap gates to make the interactions temporarily local in oder to apply the gate and perform the truncation, which can increase the entanglement a lot and therefore become quite expensive.
The two methods should match each other, given enough computational resources (big enough bond dimensions) and taking a limit of small Trotter steps. I couldn’t really give you an exact procedure for when they would match, you would need to run them both carefully and make sure they are each converged and then compare them against each other.
It should be no problem using mixed states with TDVP, see for example the discussion here: Whether TDVP can be used to calculate lindblad master equation. Basically you can just view the density matrix as a state living in a doubled Hilbert space, and make sure you define your mixed state evolution properly.