I've been wondering about the Sorting process. Specifically I've been wondering about the requisites for being sorted in the Slytherin house. Until OotP sorting seemed to be just based on character traits, the emphasis was definitely on the student's qualities. However that changed with the fifth book, at least if the Sorting Hat Song is credible. While Ravenclaw, and Gryffindor were still more or less characterized by the same criteria (that is the brave, bold, daring etc for Gryffindor, the most intelligent for Ravenclaw), Hufflepuff's founder was basically characterized as having no special character requirements, since she'd teach "the rest" when previously Hufflepuffs were characterized as just, loyal, true, patient, hard-workers etc. and the most drastic shift of emphasis was in the Slytherin description, where cunning was still mentioned, but the focus was on "pure ancestry" which the Hat never mentioned in the two other songs, IIRC. In them it was cunning, ambition, and ends over means approach, hunger for power that distinguished the Slytherin house.
I can sort of reconcile the original intentions of the Hufflepuff founder, as described by the Hat, with what the later Hufflepuff characteristics, as described by the Hat, are. However, according to the Hat in OotP, Salazar Slytherin intended his house to be "pure-blood" only. Yet we also know that the Hat does not sort only pure-blood wizards into Slytherin: Tom Riddle/Voldemort had a Muggle father after all, and it considered Harry for Slytherin, who, though not Muggle-born himself, had a Muggle-born mother, which I doubt Salazar Slytherin would have considered "purest ancestry," so the Hat doesn't seem too concerned about Salazar Slytherin's "pure-blood" agenda when considering potential candidates. And I don't see how it could be, after all since Hogwarts admits Muggle-born students, there have to be those among them who are cunning and ambitious etc and generally fit best with Slytherin, it certainly would be a problem, if in the current set-up they would be sorted into Hufflepuff, because the Hufflepuff founder intended once to accept "the rest" (as they might have been way back around Salazar Slytherin's and Helga Hufflepuff's time). Yet it can't be easy for Muggle-born Slytherins, considering the history and the founder of their house and some of its current members.
So what are the opinions on this problem? Are Muggle-born Slytherins very rare exceptions (I don't remember any besides Tom Riddle), and generally only pure-blood students are sorted into Slytherin? How much is the Sorting Hat bound by the initial founders, who created it for sorting, and their intentions for their respective houses anyway?
Now, as a caveat, I've only read the books once, and I'm not involved in the fandom or on any discussion lists, so this may be a well-trodden area for HP fans -- in that case maybe someone could point me into the direction of essays or previous discussion about this -- or I might have missed/forgotten/overlooked some further explanation in the books. But I'm interested in theories about how this works.