We have an untagged tag, which I think should not exist. What do people think? If we DO decide to remove the tag, is there any way to prevent it from showing up again?
The untagged tag is created by the system automatically typically when the Moderators delete tags from the system. Questions using that tag may end up without any tags at all. That requires a bit of cleanup.
Untagged questions should be given proper tags. That's how you get rid of the 'untagged' label; search for them and tag them properly.