1920x1080 works fine in OS 9 and Mac mini both, tested in all models personally except 1.33GHz. Have also used both 1920x1200 and 1600x1200, as well, although the former is tricky with some monitors/mini models.
The Mac mini G4 shouldn't crash unless it's software incompatibility somewhere with Mac OS 9.2.2 specifically, such as a number of very old applications crashing during "Quit". The only documented crash specifically tied to the Mac mini G4 that has been documented in all these years seems to be when software tries to access and/or make use of a microphone, such as for audio recording, at least AFAIK. So if your Mac mini is crashing outside these scenarios, then there is an issue worth looking into: RAM, CPU, GPU, temperatures or whatever.
No issues hotswapping mouse/keyboard with any of my minis, ever, both manually and via an ATEN KVM. Without the KVM, though, I get glitchy audio if my USB speakers are swapped, usually.
Audio is known to be problematic with this effort, so using an USB adapter for audio sockets/ports fixes all the known audio issues.
Unmoving mouse cursor is at times indeed also a thing, moving the mouse when booting or not is seemingly unrelated to the odds of it happening AFAICT from years of testing that theory. IIRC v9 was supposed to fix this, but doesn't. So for convenience it's good to keep a "Restart" script tied to one of your F-keys, such as F12. You do this from the "Keyboard" Control Panel, and select an AppleScript that will do it for you, or something similar.