I really like both the rhythm and the examples of the first paragraph, but think it should be changed a little, I'm just not entirely sure how. I've tried to be as constructive as I can, given that I don't have a concrete idea on how to change it.

It currently (Rev 10) reads:

Perhaps you're a computer scientist wondering how much the Axiom of Choice matters to the theorems you study and prove. Perhaps you're a mathematician who needs a top expert to explain what "randomness" really means in computability and complexity. Perhaps you need a list of problems that can be used to show polynomial-time hardness results, to strengthen your toolkit of reductions. Or perhaps you think a problem might be open -- but might just as easily be solved -- so you'd like to ask a group of professionals what they think.

I have comments on each of the four examples in this opening paragraph:

(1) I'd guess that most computer scientists have not wondered how the Axiom of Choice matters to CS; most probably just think that it doesn't. The fact that we got a really interesting answer to the contrary is a success story for the site. One of the things I enjoy most about the site is that I have learned interesting answers to questions I never would have thought to ask, not because they are not in my area, but because I always assumed the answer was trivial or never had occasion to think about them.

(2) Are there any non-CS-mathematicians who read SIGACT News? (Indeed, even the OP for that question is chair of his CS department, in addition to a joint appointment in the Math Dept.) Laurent Beinvenu's answer to that question was great, but I think we need to rephrase how we're selling it in this opening paragraph. I'm not sure the following is a good way, but I'll just note that his answer is like a short tutorial/primer on ML-randomness (basically all it lacks is the most recent results), something which might appear in a publication like SIGACT News.

Note that the amount of effort to write such a thing for such a publication is much greater than on this site, and would take several months to go through the publication pipeline, if it happened at all. Here we got a direct line to the source, and we got just the part of the tutorial we wanted, right away.

(3) A list of problems that can be used to show polynomial-time hardness results seems like a great resource for graduate students and people entering that area of research fresh, but I imagine experts in that area are familiar enough with the literature to have produced much of the entire list on their own. I think it's important that the paragraph clearly recognizes the correct audience for such a question, as is already done in the first two examples.

Again, maybe (but I'm not sure) a good way to think about it is that such a list is like part of a short tutorial that might appear in a publication like SIGACT News, but with a much lower barrier to creation (but still high quality).

(4) The last example is great. It's universal: it applies to graduate students and experts, everyone has had an experience where they wished they could do exactly this. And on this site, you can. It's a great pre-punch-line for this paragraph (the punchline, of course, being "Visit tcs.se" on the next line).