Now, this raises all sorts of questions again about MCLR. I programmed it with MCLR=ON so there is no way it should have ran when there was no power at that pin, yet it did run for a short time. What's up with that? That pin will drive me nuts yet since it does not seem to behave the way everyone here says it should behave.

NEVER leave the pin floating... if you have enabled MCLR, it's either tied to Vdd (either directly or via a Resistor), or it's tied to an external Reset chip (see answer below).

I still think I need to know how I incorporate some kind of reset in the program just in case it does crash. How do I do that?

In 99.9999% of all cases WDT will handle it for you if you have it enabled (see Datasheet). WDT is another configuration word setting like MCLR. If this is not good enough for you, then I refer you to a device like a MAX705, 706, 707, 708, DS1221 etc etc