I think there are distinctions that should be clarified:
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Preferred frame: if the Universe has a preferred frame, depending on the particular way the preferred frame is introduced into the theory, we can have deviations from both special relativity and general relativity. We have looked for this effect with precision measurements in e.g. Solar System tests and other astronomical bodies (e.g. orbiting pulsars). So far, we have found no evidence of a preferred frame, but we have not given up and are continuing to look.
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Lorentz-Aether, or Neo-Lorentzian theory: this is what @stcordova claims he adheres to. This is a subset of “preferred frame” theories. This particular preferred frame theory does not change any predictions of special relativity, i.e. it cannot be distinguished with special relativity with any experiments.
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Theories with preferred frames vs solutions with preferred frames: For example, general relativity does not have preferred frames build into its theory, but solutions of the general relativistic equations can output spacetimes with preferred frames. For example, the FLRW spacetime in cosmology has a preferred frame, the one that is expanding along with the Universe. What is interesting in physics are theories with preferred frames, not specific spacetime solutions with preferred frames. To @Jim: a-priori, the CMB is a preferred frame in the second sense: it is a preferred frame of a particular solution, not the theory itself. I don’t find these to be very interesting, but I can see how some people might find these kinds of preferred frames to be interesting theologically.
This is a hard question, and depends on what you exactly mean. Local Lorentz invariance does not grant any sort of invariance in GR in the sense that the generic GR spacetimes are not Lorentz invariance. However, as a tensor theory, GR is Lorentz covariant, which is a completely different beast from the Lorentz invariance of SR.