Re: anti-particles "moving backwards in time" ?
The idea of antiparticles being particles travelling backwards in time is built into Feynman diagrams. (Let me quote Wikipedia, "Thus, antiparticles are represented as moving backward along the time axis in Feynman diagrams.") So, yes, when an anti-neutrino arrives at a beta decay it is arriving from the future. Whether this is true in reality or a convenient fiction is another matter, but it's consistent with relativity where time is only a coordinate and makes the maths work.
Where Bob goes wrong is speculating that antimatter isn't created. The semi-rigorous take on this is that anti-matter was created but went backwards in time. So the Big Bang spawned two universe one with positive and one with negative time.