Well, this really isn't the purpose of Science Buddies, but in the interests of being helpful, your equations look pretty good to me (though I'll warn you that it's been a lot of years since I had a chemistry class). The only one that looks a little "fishy" is "NaOH + HCl => NaCl + H2O". I don't know whether it actually goes through an NaCl state, or if it would be more accurate to say "NaOH + HCl => Na+ + Cl- + H2O".
The NaCl is basically a catalyst, yes. Both electrodes "corrode", though that terminology is a bit inexact. It would be better to say that one electrode oxidizes and the other reduces.
No, the voltage doesn't depend on the surface area, The actual amperage in a circuit with the battery will follow Ohm's Law, but the internal resistance, and therefore the maximum amperage of the battery will, as will the total capacity of the battery. I don't know how to calculate those theoretically, so they sound like good subjects for an experiment