Re: During electrolysis of H2O, what happens to the hydrogen at the anode?

The additon of an electrolyte to water does not cause the ionization of water. The addition of the electrolyte just allows the water to conduct the current necessary to accomplish the electrolysis. It's the ions from the electrolyte that make the solution a better conductor; pure water is a poor conductor. So the answer to your first question is that the ionization of water (formation of H+ and OH-) does not happen, because the electrolyte is added. One way of describing what occurs at the electrodes during the electrolysis does involve the formation of these ions as a result of the electrolysis. In dealing with anode and cathode reactions, one must be careful to remember that in the overall process, the number of electrons lost must be equal to the number of electrons gained. In this case that number must be four at each electrode.

Thus the cathode reaction is:

    4H2O + 4e-  --->  4H + 4OH-

You can think of the process this way:
Each water molecule is pressureed (by the voltage) to accept an electron. When this happens, a H atom is released and an OH- ion is produced. This OH- ion makes the solution near the cathode basic; this can be shown with an acid base indicator.

What you have written for the anode is reasonable, except again the process must start with H2O not H+ and OH- AND you must get 4H+ not 2H+ in the first step. Notice that you used 4e- here.

The four OH- from the cathode eventually meet up with the 4H+ from the anode to remake 4 water molecules.

Overall 4 water molecules were invested at the cathode and four at the anode. Of these eight, two were regenerated from 2H₂O₂, two were decomposed to give 2H₂ + O₂ and four more are regenerated from.
4H+ + 4OH- ---> 4H₂O

I hope that this helps to answer your questions.