Can Data Explorer do those things? For example:
Search all of the questions which have the only tag of the particular tag
Search all of the questions which have the particular tag(s) but haven't the particular tag(s)
Yes, see Recent questions with only one tag . It actually returns all questions with the given tag as the only tag, with most recent first.
This is easier to do with the built-in search. Precede the undesired tags with
-. For example, you can search for
[abstract-algebra] [polynomials] -[homework] -[group-theory]