I don't see any need for this, and I don't see how you could define "meets all other site criteria" and "a genuine question."
Both processes - downvoting and voting to close - are in place for good reasons. Downvoting to indicate that people don't think the question is good, and voting to close to indicate that it needs substantial work or doesn't belong here at all.
Your first paragraph posits a case where the question is bad enough to be downvoted and to have some close votes thrown on it, but somehow it'll be found to be worthwhile enough to give someone a badge for it. Maybe there's the odd question that fits that description, but you'd have to point out a bunch to convince me.
Similarly for your second situation, you'd have to cite some examples of answers that are accurate and supported yet received downvotes due to unpopularity.
And for both examples, you'd have to cite examples where the net votes are negative, not just an answer that's not as highly upvoted as a less-correct one and not answers that received a few downvotes along with a bunch of upvotes. Again, I think there are very few of these, so what's the need for a badge?