The Wikipedia article is confusing the Higgs field with the inflaton, which is not allowed, since it is not a Higgs VEV that causes inflation in current theories. This should clear up the questions.
- If the Higgs is the inflaton, it has a nonthermal, nonequilbrium value, which slowly oozes to the vacuum value and this is while the universe inflates. This process requires that the Higgs potential at large values of the field is abnormally flat, so it works as an inflaton, and this is not supported by current understanding. Once inflation is done the Higgs oscillates around the minimum (our vacuum value) and emits standard model stuff, then settles down to a stable vacuum value. From this point on, it is nonthermal vacuum field, and doesn't change as the universe expands. This is not the modern picture of inflation, where the inflaton is separate field from the Higgs.
- If the Higgs is an inflaton, this is a false claim. A big Higgs VEV means big masses for the SM particles. In the standard inflation ideas, the Higgs is not the inflaton, the inflaton is a different scalar that is slowly sinking to zero causing inflation, and at the end, it pumps up other scalars and these decay into thermal stuff, all at energies above the Higgs scale. Then the Higgs in the early universe starts out thermal and random, with a temperature higher than the scale at which it breaks symmetry, so it's average value is zero. In this picture, random thermal Higgs, the particles are effectively massless (although by construction these temperatures that randomize the Higgs are so high, that the particles are whizzing around so fast that their Higgs-derived mass is negligible anyway). The average thermal value of the Higgs has a phase transition to zero at extremely high temperatures, but the article is confusing the no-mass high temperature Higgs with the high VEV Higgs required when Higgs is made the inflaton. In the second case, the masses of the SM particles are not zero, since there is more and more Higgsing at larger Higgs VEV.
- The interaction with photons is indirect, through gravity. An inflaton makes a cosmological constant, which makes gravity repulsive, and then the photons follow along. This is not the right picture, though. There are no photons during inflation, the universe is just inflaton VEV. The photons come later, when the inflaton decays to standard model stuff at the end of inflation.
- stationary here is defined relative to the decaying nucleus, and this is not a correct statement--- the electrons are produced left handed at "t=0", but the mass of the electron flips their helicity constantly. The neutrinos are the ones that stay with one helicity, since they have no partner.
- this is a very interesting point, and one that should be asked as a separate question. The traditional answer is that a charged particle can be massless so long as the chiral symmetry is preserved at long distances in the theory, but it is intuitively puzzling to have a charged massless particle, because, as you say, the field has mass, so how could you get a long-range electrostatic field from a massless particle? I don't know the answer immediately.