I think most of the heavy users of the site would be very happy to see questions asked out of pure idle curiosity: pure idle curiosity is one of the nobler motives in mathematical research, so far as I am concerned. It sure beats the hell out of "I was assigned this question so I'll try to get it answered online."
I can see that you are mostly questioning the desirability of sort of half-baked questions along those lines. To that I would say: it takes a certain amount of work to post a question on math.se -- probably at least a minute or two of typing, right? Before you spend a minute or two typing in any question, spend a minute or two -- heck, why not five? -- thinking about it. To an extent this is a bit counter to ordinary psychology: in many aspects of life, if a curiosity pops into your head, then since you're curious you necessarily don't know the answer, so to get it you need to look somewhere outside of your own head. But it is not necessarily so in mathematics. For instance, recently on this site someone asked a question of the form "Are there fields which have property X?" And I gave "the general answer": "A field has property X if and only if...." Another (much younger) answerer commented that he had looked through my field theory notes and not seen that theorem before. My reply: neither had I! Remarkably, in mathematics you can use what you already to know to discover entirely new things. Idle curiosity together with a willingness to try to answer your own questions: well, you can make a research career out of that.
Of course many times you simply won't be able to answer your question in five minutes or so. But you'll probably be able to come up with something to say about it, and I think you should include that something in your question to show your engagement in the process. For instance, if you want to ask about the number of subgroups of $S_n$, you could observe that by Cayley's Theorem every group of order at most $n$ occurs up to isomorphism as a subgroup of $S_n$, so very roughly speaking the answer is: "a lot". That's not the deepest observation in the world, but it shows we're thinking, right?
About "stupid questions"...Teachers are supposed to say that there are no stupid questions, but unfortunately that's not true in all contexts. There are, sadly, some questions which when asked make those who hear it think less of the questioner. I think a good modern-day answer to this is that no one can see your stupid googles (well, no one you know, anyway). You really can answer most of these questions simply by googling them, even when you don't know exactly the right keywords (i.e., google is miraculously good at what it does). Just now I googled "subgroups divide the order of their supergroup". "Supergroup" isn't really the right word, so I was curious to see whether that would get you to Lagrange's Theorem. It sure does.
