As proof of concept, I am trying to accomplish one of two things:
1) Wire the pi such that it will automatically boot up upon shutdown
2) Turn the power to the Pi on from a separate Pi. This seems as though it would be possible by shorting either the "RUN" pins or pins 5 and 6.
I don't know too much about circuits and would really appreciate any help that I can get.
You'd need to edit the Linux (if that's even possible) so that the shutdown command becomes a reboot command. That creates one unforeseen problem though: aside from unplugging the Pi, you won't be able to shut it down unless you create a separate shutdown command of some kind. Otherwise, every time you go to shutdown, you'll get a restart.
WRT powering the pi from GPIO activity, that promises to be more of a challenge. Since you'd need the Pi to be on to monitor the activity of the GPIO pins, you're caught in a logical loop - it needs to be on to turn it on. if you connect the grounds together, you should be able to make the pi shut down by sending power from one pin on Pi A to a pin on Pi B. (in that case, just monitor the GPIO pin and initiate the shutdown command when input is detected).
If you have the two Pis networked you could ping PI1 from PI2 and if PI1 fails reset it via a relay on PI2.