I like the criteria in the question. To reiterate:
To be valid Purim Torah on Mi Yodeya, a question must have at least one of the following three components:
- A mis-application of a Torah principle
- A mis-interpretation of a Jewish text, be it Tanach, Talmud, or some later writing
- A "distinctly Torah style (e.g. Gemara/Rashi/Tosafot-formatted) analysis of some secular issue."
I suggest that questions that fail this check be closed as off-topic. Rather than trying to manage a fourth custom reason (which we only need for a few weeks a year), just choose "other" and type a comment. Or if some other close reason applies, like "unclear", use that. The point is to get them closed, and the community can help with that. (That said, my proposal allows mods to act unilaterally too.)
Then, in the spirit of DoubleAA's answer, questions that aren't fixed should be deleted. My proposal:
If it's downvoted and not fixed within 48 hours (excluding Shabbat), a mod can delete when it's noticed. (If it has no answers the Roomba will delete it eventually, but this expedites cleanup of bad questions.) I'm allowing more time than DoubleAA did because they're already closed, which limits the damage, but if people think 24 hours is sufficient I can go along with that.
If it's not downvoted and not fixed by the end of the Purim Torah season, mods should delete it before the mass closures. (If it's bad, people should express that with votes. If it's just "meh", there's no need to take extraordinary action.)
If it gets fixed and reopened, it becomes a normal Purim Torah question -- win!
Of course, the community is always free to delete posts outside of these parameters, should that seem warranted.
A comment asks how the presence of answers, upvoted answers, or upvoted self-answers would modify this. My inclination is to say it doesn't matter; if the question should be closed under these rules then people should close it and not answer it. We might lose a few good answers, but those answers stand on no foundation, so how valuable are they for the site? (If they're that good, maybe they can be applied to an on-topic question, which anybody interested could ask.)
The guidance from SE on deleting closed questions includes this:
Off Topic: Almost always delete it.
Off topic usually says "This shouldn't have been posted here in the first place." So, for anything not about topic [x] or the off-topic reasons, just let it burn!
If that's the guidance for regular questions, I would expect the approach to be even stricter for things like Purim Torah that are tangential to the main purpose of the site.
(Note: I don't think we wholly follow that guidance for other questions, nor am I saying we should. This post is specifically about Purim Torah -- which, I should note, I love when it's well-done or at least not so far afield that I wonder why it's here.)