What are the various ways that a question can end up tagged as untagged?
Tags that only exist on one question are automatically removed after 6 months (12 months on meta sites), unless the tag has a tag wiki. As well, tags are sometimes burninated through Meta discussions. If the tag was the only one on a question and not all the questions were edited before the tag was removed, these actions will leave the question untagged.
Additionally, if a question is migrated from another site, and none of that question's tags exist on the destination site, the untagged tag will be applied. Ordinarily, such migrations are blocked, but moderators of the origin site have the ability to bypass this check, and the check is also bypassed if the question uses a tag that is considered "intrinsic" to the destination (e.g. graphics where Graphic Design is the destination). (This does not apply if the destination site is a meta site, in which case it will get the tag discussion.)
Can the untagged tag be added to questions, or must it be removed when another tag is added?
The untagged tag is blacklisted network-wide, so you cannot add it to any question, and if you edit a question that uses it, you'll be prompted to remove it.
You should periodically check for untagged questions and retag them (or close/delete them if necessary).