1. We haven't tried to play solo for any length of time (just enough to get somewhere safe to shut down the server), but errors have immediately stopped when someone logs out, even if it's not the player who is having the error. It had seemed like it was somehow related to who had joined first, but there has been a single instance where the host (in non-dedi mode) received it (compared to 10+ times where it was the second person or non-host). That was fixed by the other person logging out, so it's possible this can only happen with multiple players.
2. No, though we only replaced them in a non-error state (when the errors are flowing we can just struggle to exit out). We did get errors on new items pulled from the creative menu.
3/4. No custom UI's or other mods, just the version provided by the launcher and the dedicated server from github.
I'll try to gather the logs and save game and send them over. The errors aren't reliable though, so it won't be like you can just load it up and swap around to get it to occur.