I switched cables on my monitor to a displayport to DVI cable, so that I'm outputting on a different port on the graphics card. I still saw the static, though it was more frequent once I did that. I switched the monitor and that fixed the problem. I no longer see any static now that I've switched out the monitor.
At this point all of the original issues are gone, save that the BIOS start up screen still looks corrupted. The computer sleeps, the computer starts up without problems, and the monitor no longer flickers. This is great because it means I won't have to spend any more money on it for now.
As an aside, when I bought the monitor I was using, 10 years ago, the guy at Staples tried to sell me on a lifetime warranty from Staples for $50-100. What would they do if I had bought the warranty and took back the monitor today? That model has been long since discontinued and you can hardly buy a monitor that is 1920x1200 with 500 nits brightness anymore.