Update;

Did all teh work outlined above. Filled, ran, and then drained the system about six times before it stopped coming out rust colored with sparkles in it. Replaced the thermostat, replaced radiator cap, refilled system with 50/50 and idled up to 200*. Aux fan came on, everything remained around 200. There was still a fair amount of pressure in the system, temps were consistent with previous checks. After shutting off, still got bubbles in the overflow. I've been limiting use to short trips within a few blocks of home. Everything above has been repeating. Took it to work today, which is a bit further than the few blocks, and temp climbed up to about 220. Turned on defroster to keep it in check, which allowed me to complete the drive.

So it seems I haven't actually fixed anything. It still gets overly hot on occasion and still blows bubbles back into the overflow after shut down. What's next?