Previous comment about SR, although GR is a natural extension of SR for non-inertial reference frames.
> Moreover, interpretation of Michelson-Morley experiment as «it proves that physical vacuum does not exist» is frivolous.
The interpretation of it is that it proves the constancy of the speed of light. This is the case whether SR is assumed or Lorentz ether theory. The two theories result in identical predictions, except LET assumes the existence of an ether but without any way of detecting it.
> So, Michelson-Morley experiment confirmed that physical vacuum is dragged with Earth, and then Michelson–Gale–Pearson experiment confirmed that it is not completely dragged by Earth.
Michelson-Morley rules out a stationary ether, at least without length contraction which Lorentz added to attempt to repair the theory. Michelson-Gale-Pearson rules out ether drag. SR and LET are compatible with both experiments. Whether or not the ether is real is philosophical quibbling without a method for separating LET from SR.
> * the existence of gravitational waves in physical vacuum is proved by LIGO/Virgo interferometers.
This is one of the predictions of GR, so I'm not sure what your point here is.
> Thus, today we have Higgs field, which is presented at every point, which can be used as 0 point, but we cannot detect it. However, we can use CMB (light from distant galaxies with red shift z=1000) as 0 point.
> IMHO, relativity is not necessary at this point.
The Higgs field, just like other fields in quantum field theory, is a relativistic field; its Lagrangian is Lorentz-invariant.