It seems to me that the criteria used to award the "tumbleweed" badge should be usable as a search filter. It would serve as a second chance mechanism for questions that fell through the cracks.
Er.. you can already do this. Are you not familiar with advanced search? There has been a ton of Q&A on meta about it.
(the only part you can't do is select age of question, but given that you can already query for a minimum vote and view threshold, and specify zero answers, that seems irrelevant)
This could be generalized to search for questions or answers which have earned the user a particular badge, e.g.
badge:tumbleweed. Although reputation earned for questions and answers is readily available from the DB (since this data is shown in the user's profile under "Reputation"), it is not clear whether the DB stores how a badge was earned.