Have you tried SSH-ing into the system when it's in the bad state to see if you can diagnose the problem? You might be able to see if any displays are being detected at all in the problematic state. Part of me wonders, though is not certain, if the switch is somehow providing an inconsistent display name that confuses the system, though this is just a hunch - I have no idea what I'm talking about, to be frank.
Also, try switching TTYs and seeing if those show up.
It depends. Sometimes I shut it down every night. Occasionally, I'll leave it in sleep mode for a few days.
I think the longest uptime I've had on anything I've owned is probably a month or so on a Raspberry Pi 4 server I used to have running with a personal Mediawiki instance (I still have the Pi, but if I ran a server in my dorm, I have the feeling someone might come to bite off my hand).