Environmentally induced decoherence makes wave function collapse unnecessary. But the environment, usually taken to be some heat bath, introduces a preferred frame. (That in which the total (spatial) momentum vanishes.) So, doesn't then the decoherence time depend on the motion of the prepared state relative to the environment? And, doesn't the ultimate environment, all particles in the universe, introduce a preferred frame into quantum mechanics in the sense that the decoherence time is relative to this frame? And would this be measureable, at least in principle? I.e. I could go into a frame with high boost relative to the CMB restframe and notice that the decoherence time changes?
As far as I know, that's still a conjecture. I might be wrong about that, though.
It's the rest frame of the system + reservoir, but it is not "preferred" in the sense that it violates Lorentz invariance.
Only the part of the environment that is directly interacting with your system. If your reservoir is a collection of matter interacting with the system by the exchange of blackbody radiation photons, then the decoherence time will depend on how often the system interacts with one of those photons.
What make you think that the universe is finite?
All decoherence times can be calculated from the decoherence times of the same system in any other inertial frame. The rest frame of the observable universe is in no way "preferred".
That decoherence is affected by time dilation like everything else? Yes.
Long story short: you probably aren't going to be able to use decoherence to find some universal preferred frame and disprove special relativity.