I've heard it's not the electrolysis of water creating hydrogen (straight electrolysis requires much higher temps), but reaction with the Zn (oxidation), which releases hydrogen. Yeah, it releases hydrogen, which is vented off (and can explode, as we've seen), but it's better than letting the rods melt, and possibly melt out of the containment vessel.
As far as the minerals in seawater goes, water itself is a neutron moderator, which can slow the neutrons down getting the reaction back to criticality. I've heard the control rods Scrammed in were enough to ensure sub-critical mass, but the boric acid was added just as a safe measure, though they think it's unnecessary.
If they can keep up the cooling for at least a week on reactors 1-3, the decay heat is suppose to drop to 50% or so. After that, it'll requires 2 or 3 more YEARS of cooling until the core can be removed (I think).