The popular definition of the rhetorical question seems to be: 'a question that does not expect an answer.' And, like most popular definitions, it is quite wrong. In fact, the rhetorical question goes further: it is a type of question that demands a response. The point of the rhetorical question is that it has a point. It expects an answer, often a quite specific one: 'no', for example, or 'yes'; but it may demand a response of a different order: an emotional reaction, an intellectual move, or an act. The undermining of the term in popular speech is a corollary to the decline of rhetoric more generally speaking. If the RQ (I adopt the abbreviation in preference to the use of the pernickety classical term erotema) really were a question that does not expect a response, then this subtle rhetorical technique has somehow become indistinguishable from base sarcasm. 'What are you, some sort of idiot?' is not a rhetorical question; it is a sarcastic one (as is, for that matter 'What am I, some sort of idiot?'). Ditto 'Why don't you just shut up?'
RQs do not always expect a specific answer, of course. In fact, the more subtle deployment of the technique is in the formulation of a question that gets at some point without coming to the point. This is where the RQ brushes up against the paradox.
RQs always had a better time of it in classical oratory, since both Greek and Latin grammars facilitate a basic distinction between questions that expect the answer 'yes' and questions that expect the answer 'no'. English gets around this in various clumsy auxiliary and adverbial ways ('Surely you don't agree with him?' [No!] or 'You do agree with me, don't you?' [Yes!]); but it foregoes the elegance of a wittily deployed 'num...?' or the straight-to-the-point effectiveness of a trenchant 'nonne...?' (starting a question with 'NOT NO' drops a pretty big hint on where you're going with it). Mind you, subtlety was not necessarily what was sought after in classical forensic oratory: Cicero's speeches are full of pretty outrageous assumptions and beggings-of-the-question packaged up in RQs. Starting a speech against a man suspected of sedition with: 'How long do we have to put up with this shit? How long are you going to keep playing us for fools, you crazy bastard?' (In Catilinam I; I paraphrase) is not a particularly even-handed way of going about things. In fact, it smacks less of the rhetorical question and more of the 'Have you stopped beating your wife yet?' double-bind.
All of this suggests that RQs are merely statements (or accusations, orders, requests, exhortations, etc.) masquerading as questions. But that doesn't tell the whole story. RQs are, it is true, basically illocutionary in that they are formulated to achieve something in their very utterance. The masquerade is not mere sophistry, since the reformulation of an utterance as a question actually changes the content of the utterance.
It makes it more ambiguous, for one thing. 'Will no-one rid me of this meddlesome priest?' might have been just a casual expression of exasperation (with an undertone of toys-out-of-the-pram 'Who's the King here, anyway?'), but it was taken as an order, and the speaker lived to regret it. That was unintentional (if it's true), but this ambiguity can also be a feature of self-aware speech or writing. The RQ does not only add force to certainties; it can also be used as a matter of principle in sceptical enquiry. 'What do I know?' is itself a rhetorical question, because it exhorts us to keep looking. That's why Montaigne adopted it as his device.
So leading questions are all well and good, but what about the more subtle variety of rhetorical question, the one that doesn't really want to convince you of its point, but rather, to make you think about the implications of the question? The RQ in this mode is more poetic, too. What about the 'ubi sunt...?' motif, a once popular poetic form that is structured around the repetition of a rhetorical question ('Where are they now?')? Alternatively, what about something like this: 'And did those feet in ancient time / Walk upon England's mountains green?' No, they didn't, of course, but what does it mean to have asked the question? Or else: 'All the lonely people / Where do they all come from?' The question does not make much sense; and no answer presents itself. Nevertheless, it is moving somehow, in a way that the same sentiment formulated differently would not be.
Another variation, and possibly a vice in my own writing, is the asking of a question followed by the supplying of an immediate negative response (anthypophora if you like). The speaker here can end up taking on a hectoring tone, bludgeoning his audience into submission by the sheer force of his own monomaniacal irrelevancy. Or he may proceed, more Socratico by burying whatever claim he is really making under a morass of banal and idiotic questions that really have nothing at all to do with the argument. Schopenhauer recommends using this technique in The Art of Always Being Right.
The RQ is used a lot as an act of imposture. If you're making some outrageous, counter-intuitive claim, be sure to formulate it as a question. That way an absurdity might pass by unnoticed. Slavoj Zizek does this a lot, and it works quite well. But isn't that precisely because the logic of late capitalism forces us to consume pathologically while dissimulating the disavowed truth of our own desire, and is it not therefore true that the symbolic order requires us to displace our beliefs onto it in order that ideologies of exploitation might sustain their hegemony? The answer to these kinds of RQ tends to be: 'Maybe...but probably not.'
There's also the use of the RQ to evade responsibility for one's own utterances. But whose fault is that?