I think there are two aspects to bringing in questions from elsewhere: impact on the source, and impact on the community.
Impact on the source
Stack Exchange welcomes questions from anywhere, and in a blog post I can't find now, Jeff Atwood or Joel Spolsky (IIRC) explicitly suggested the following strategy for promoting SE sites: find a question "in the wild", e.g. somebody asking on Twitter, come to SE, ask it, and respond to that person with a link to the answer. Therefore, from SE's perspective it's completely ok to ask questions that didn't originate with us.
However, both SE and our Jewish values forbid plagiarism. You should either ask the question in your own words or quote it with attribution. The latter should be reserved for questions that are a relatively small part of a larger whole; if a textbook were to present material and end each chapter with a few questions for further thought then I wouldn't see a problem with quoting those questions (with attribution), but if a publication consisted entirely of such questions, I would be hesitant to bring more than a very few of them here in that way.
Re-asking a question in your own words is generally fine. It's polite to acknowledge the source if the question is particularly insightful or well-sourced -- give credit where it's due -- but if someone at shul asked you a run-of-the-mill question and you brought it here, I wouldn't expect you to name the person. I've described ends of a spectrum and most cases will be in the fuzzy middle ground, so use your best judgement. For what it's worth, I have re-asked questions from elsewhere to SE and also from SE to a blog (where my goal was to post my answers, but I needed questions to hang them on).
Impact on the community
We should strive to ask questions that will bring some value to Mi Yodeya. Riddles fall below this threshold (thanks Shokhet for the link), but real questions (that satisfy our scope requirements) are welcome. Be mindful, however, that there are only so many slots on the front page, which may be the only page many people look at, so try to spread them out. We want good questions and we also want a good variety of good questions. The front page that naturally evolves contains questions on a variety of subjects, from a variety of people, requiring different levels of expertise to answer. Please respect that and help keep the site engaging for all.
