The fist 3 minutes is a misinterpretation of the Einstein-Podolsky-Rosen Paradox [1] and the Bell's inequality experiments [2]. (The page he shows is [3].) The experiment don't refute Special Relativity, it only "almost" [0] confirms the current formulation of Quantum Mechanics.
The first theory that joined QM and SR was the Dirac equation for the electron [4].
Currently all the "serious" equations to describe particles mix QM and SR. For more details, see [5]. It's difficult to understand, but each column in the table describe a different kind of particle. All the Greek subindex and superindex means that each equation is compatible with SR. (It's a very nice notation when you are use to it.)
All the calculations in the LHC and other particles accelerators relays in this equations because the particles go extremely fast, so they are very well tested.
Then the video gets worse, but I don't have time to refute each sentence.
[0] I wrote "almost" because I think the "no loopholes" claim is an exaggeration. It closes all the weird and very weird loopholes, but there may be some extremely weird loopholes that no one has imagined. If this were Mythbusters, I think this experiment would be more than enough to put a "confirmed" sign.
[1] https://en.wikipedia.org/wiki/EPR_paradox
[2] https://en.wikipedia.org/wiki/Bell_test_experiments
[3] http://arstechnica.com/science/2015/08/experiment-confirms-t...
[4] https://en.wikipedia.org/wiki/Dirac_equation
[5] https://en.wikipedia.org/wiki/Relativistic_wave_equations
