This answer right now mostly addresses the question of "how these questions should be asked", I will edit it later with a response to "how should they be answered" once I have more time to write. I believe that the easiest way to avoid having issues over things like this is fourfold:
1) Don't make universal claims. One of the first things your professor tells you in your year one composition class (or at least the first day of debate class) when you learn about persuasive writing is "don't make a universal claim unless you can back it up." Making claims like "all x believe y" when you don't have any sort of data just hurts your argument, detracts from what you're saying, and makes it easy to criticize. Like you said, finding one x that doesn't believe y makes what the person said untrue. Just don't make universal claims, e.g. "all people who like orange juice are impious," or whatever it may be.
2) Link to some sort of support for the claim that "some x believe y". Yes, everyone knows that there are tons of issues with statistical polling, especially self reporting, but some sort of poll or otherwise quantified data puts even just a little bit of weight behind what the person is saying. Instead of saying "all people who like orange juice are impious," you could instead say "This study showed that there is a positive correlation between people who like orange juice and impiety." At that point, nobody can come in and yell "NOT TRUE! I AM A ORANGE JUICE DRINKER AND I LOVE GOD" because a universal claim wasn't made, but more importantly if they want to disagree, they have to disagree with the actual findings, not just with someones opinion. If someone gives anecdotal evidence saying "well everyone I know who drinks orange juice are consistently late on getting to church and don't devote enough time to prayer, therefore I assume impiety is a large problem in the orange-drinking community," then it's very easy for someone to point out the issues. If instead, as I'm suggesting, they link to some sort of research showing a correlation, people who want to argue the point have to disagree with the research, which gives them less credibility than disagreeing with an opinion. To your point about "we don't need to quote a Pew study,": it does not matter if we already have a preconceived notion. Treat this like you would treat writing an essay for school. If you were in 10th grade composition class and you wrote in a paper "American culture is based on consumerism and everybody knows that this is true," without any other sort of backing up with points, don't you think your teacher would circle that in red pen and say "needs supporting argument"? If the goal is to avoid having conflict over these types of questions, saying "well there are just some generalizations about people in the world that we all know are true" doesn't cut it. Support your argument by using data and don't make universal claims, just like how we were all (just kidding, "all" is a universal claim) taught to write essays. If you (and for clarity, I'm using the royal "you", not you, OP, in particular) don't want to make the effort to have a well written and correctly presented question, then that's fine, but then you can't be upset that it is criticized. Again, I think if the question is "how should we go about asking these questions" then the answer is that there needs to be some sort of effort put into presenting the statements as more than just an opinion.
3) Tone matters more than anything else. Using provocative or grandiose language (especially when it is involved with universal claims) is going to get people angry or otherwise want to point out the flaws. Saying something like "All orange juice drinkers are disgusting pigs" is designed to get people upset. Having a purposefully antagonistic tone in your question is going to get people to argue, no matter who is right or if anyone is right.
4) Point three goes into a larger point about tone: the site has a very specific set of rules on how to communicate nicely and anything that violates those rules is subject to either be deleted by a moderator or otherwise changed to fit the tone of language that's asked for. It doesn't matter how much you hate something that somebody does, or a group of people for what they believe, or a specific ideology or opinion in general. This site has rules against being hurtful and antagonizing on purpose and any questions that do so are violating the rules, it is that simple. So, how this partly answers the question of "how should we treat questions about heterogenous groups," is "follow the rules of how to ask questions on the site and there shouldn't be a problem."
When you have a question that calls a group of people pigs, it is going to get negative attention and it will either be closed or edits will have to be made to follow the rules of the site. Beyond the tone issue, the question made unsupported universal claims, generalizations, and later was edited to try and use anecdotal evidence to support the claims. That would not fly if you were writing an essay, and for people that care about the quality of this site, that will not fly here either. If you dislike a certain school of philosophy or a certain ideology in general, loading all of your questions with "I hate this line of thinking, it has ruined the world and I do not understand how a single person could believe it," is not neutral, it's pushing your own idea and falls under the category of "You should probably rephrase the question in a more neutral manner and then add your opinions as an answer." To be fair, the postmodern questions are not guilty of lack of sources and substantive links, but they are guilty of making universal claims like "Postmodernism is destroying serious philosophy," which is an actual quote from one of the posts (meta answer on the issue with our lack of nominations for moderator).
It requires more effort to supply some sort of arguments to back up why you are asking the question you want to ask, but if you care about the content and quality of this site, then that effort is something that will be necessary. Of course, this is also why we have the voting system and the collective sense of community acceptance. Sure, anybody is free to post their universal claims and provocative questions, but it is very evident how the over all user base responds to those questions by how they use the SE systems to respond to them (voting to close, downvoting, etc.). If people want to genuinely ask questions like the one's you're referencing and describing, they're going to have to put in actual effort to make them productive, well meaning, and well composed questions.