I spent more time diagnosing today. First of all, here's a more representative video of the problem. The engine is running, but the gauges are going crazy:
https://www.youtube.com/watch?v=t4urfxNYUiA
I started by checking the ground points in the front and rear trunk, and they looked perfect. I then decided to wade through my wiring diagram book to look for common ground and/or power sources for the various issues (gauges, dash, reverse light, brake light, cruise control, starter/key switch.) After about an hour of flipping between pages, I couldn't find any commonalities.
So I decided to take a different tact by removing the fuses for the problem components. Dash lights, reverse lights, brake lights did address the issue. But when I removed the fuse for the the gauges, it started immediately every time. Brakes and reverse lights started functioning properly as well.
So the cause of the problem seems to be related to the gauge cluster causing electrical problems for other systems.