I understand the leveling effect at the acid end of the scale. pH is a measure of hydronium concentration and there's 55.5 M water in a sample of water. So no matter how much of even the strongest acid you add, you can't get more hydronium than this. A hydronium concentration of 55.5 M corresponds to pH = -1.74. This is the lower limit of pH in aqueous solution. Usually this effect is laid at the feet of hydronium having a pKa of -1.74, but explaining it that way is less intuitive, I think.
Q1) Is it a coincidence that the pKa of hydronium is -1.74? Or does that follow from the concentration of pure water?

The usual explanation for the leveling effect at the high end of the pH scale is that you can't have anything more basic than hydroxide in aqueous solution, since it would just strip a proton from the water to make hydroxide. Water's pKa is 15.7, so aqueous solution can't have a pH higher than this. This is a really lousy explanation, since we're talking about the HYDRONIUM concentration, not hydroxide.
Q2) Why can I not, with a strong enough base, remove ALL the hydronium from solution and convert ALL the water to hydroxide? If I did this, the pH would go to infinity, not 15.7. In fact, a pH of 15.7 corresponds to a hydronium concentration of 1.8x10-16 M. What's so special about this concentration?

I've looked about quite a bit for an explanation, but all I get is the same tired rehash of the business about the pKa of water. I'm really hoping for an explanation of the phenomenon in terms of how the molecules are interacting which explains why you CAN'T lower the hydronium concentration below 1.8x10-16 M.
Thanks