The Sorting Hat is a horcrux. It HAS to be - or some very close analog thereof.
Why else would Riddle want to turn all the other Founders' artifacts into horcruxes? It wasn't just claiming territory by pissing on it. The Sorting Hat was already a horcrux. It dictated the path of every single student in that school and the students who came back as faculty. Hence, the Sorting Hat even controlled the faculty. And it isn't hard to imagine that where the Sorting Hat places you eventually dictates the KIND of magic you learn and/or specialize in. I'll have to expand on this later but what an INTRIGUING idea.
So the Sorting Hat has a LOT of power, even moreso, in a way, than the Headmaster does. The hat stays in the Headmaster's office and apparently, aside from the Sorting Ceremony and Fawks taking it to Harry in the Chamber of Secrets it never leaves it.
And can you imagine if there were books in the restricted section that told you how to create a horcrux? You could create something that could oppose the hat, thus undermining the entire basis of how the school operates.
Now THAT sounds like a good reason for trying to turn the Founders' Artifacts into horcuxes. Riddle wanted to undermine the hat and so undermine the school. He wanted to take over the school so he wanted horcruxes that could over power the hat and create a new basis from which to operate.
Riddle probably first learned about horcruxes simply by taking the time to talk to (as well as study) the hat. Simple.
This doesn't answer everything. As I said, this is all off the top of my head. But why else would Riddle be so agitated about getting and keeping hold of the sword. Maybe it wasn't Fawkes that sent the sword. Maybe it was the HAT that told Fawkes to take the sword to Harry. So it might be that the hat and the sword work together somehow - a different kind of horcrux perhaps. Or perhaps Riddle never truly understood what the hat was and so mistakenly concluded it was a horcrux and hence he would need to create others to counter it. Oh Riddle still wanted to cheat death. But after that he wanted POWER. And we know that the horcruxes were magically functional not just as storing devices but in self-defense as well as in poisoning the mind. So what ELSE can a horcrux do?
OTHER NOTES
1. Allegedly, before Pottermore, it was Herpo the Fowl sometime around the dark ages I think, who created a horcrux. I never though that made since. The concept is ancient and so older than that. Another fan writer whose name escapes thought it was Egyptian which makes a lot of sense since they were greatly concerned with resurrection.
2. Hence the horcrux might be a DESECRATION of something that originally was good and while probably connected to death probably also didn't require a ritualized or non-ritualized murder to create it.
3. Students would never be allowed to learn about horcruxes not just because they are evil but because knowing about them endangers the school's power base. If you know the mechanism of house selection, you can then question it and pick it apart. If you pick it apart, then the allocation of students can be challenged and power relationships altered.
4. As I mentioned before, the horcruxes may have had another function that was never mentioned but which Riddle knew about. Dumbledore might have known about it too.
5. Bad writing aside, the Deathly Hallows are connected to the horcruxes. I just don't know how. For Rowling it was probably just a device to explain the wizard obsession with defeating Death. But I still feel like there's a connection somehow. And something makes me think Harry's invisibility cloak has something to do with it.
6. Am I the only one who thought the story of the Three Brothers was bullshit? It feels like a redaction. It's entirely too pat. It was never explained WHY Death felt it had been cheated. People cheat Death all the time and he doesn't go hunting them down. Because he knows everyone comes to him in the end. There is no reason for Death to be so ravening. Also, there is no question of ever meeting Death as an equal. You can be unafraid of it but it WILL defeat you in the end (something Gryffindors have a VERY hard time accepting I suspect. They specialize in being reckless and cheating Death. Look at the Marauders. Look at Harry.)
7. And why WOULD James not have his cloak? Why did Dumbledore have it? Ever notice that a hat, a cloak and a sword practically make a complete ensemble? Think about that...
8. Why would Riddle NOT know about the Deathly Hallows? After all, the Gaunt family ring WAS the Resurrection Stone (perhaps a fragment of the Ben Ben stone of Egypt and which was connected with Phoenixes...). And Riddle DID know about the Elder Wand. Why did he not know about the cloak? Why did he not recognize the Resurrection Stone? Why was he able to turn that stone, allegedly an artifact of Death, into a horcrux? Something is VERY fishy here. My first thought is that the Death of the Three Brothers was not the real Death at all but something else... Actually, in my Judeo-Christian worldview, the one creature I can think of feeling cheated of three souls would be Lucifer/Satan. Why? He's known as the adversary. That means he is an ACTIVE enemy of mankind. Hence every bad death is a victory for him. If he can bring about death through treachery, he will do it and feel cheated if he doesn't accomplish it. Death is a neutral force, subservient to God. The adversary is in rebellion against God and hence will lie, cheat, steal and most of all CORRUPT. What is the result of the Deathly Hallows? The eldest brother becomes an arrogant bully who is then murdered for his wand. The middle brother resurrects his love and becomes so despairing of truly being able to be with her that he kills himself. Traditionally this is a mortal and unforgiveable sin that bars you from Heaven forever (this is an outdated notion of course; I don't think that's true). So the middle brother is damned for eternity. What about the 3rd brother? He hid all his life. Only when he could no longer escape the reality of Death did he surrender the cloak to his son and meet Death as a so-called equal. His end appears to have been peaceful although to me his life is a bit dodgy. Since his end was allegedly peaceful it's possible at this point he met the real Death, not the imposter. I can't say.
9. So did Tom not know about the Deathly Hallows? Or was it that he DID know but turned away from them as an option? And again why did Dumbledore have the cloak if it could have saved the Potters' lives?
10. Oh and, why WASN'T the sword a horcrux? Probably because Dumbledore was in the office and had it hidden. Tom wouldn't have had enough power to find and acquire it.
11. And why did Godric Gryffindor have TWO artifacts in the school while the other Founders seemed to have only 1 each? EXCEPT there actually were also TWO for Salazar Slytherin: the locket and the Resurrection Stone, neither of which were in the school, unlike Godric. And that tells me that the split with Slytherin had something to do with the Deathly Hallows and perhaps horcruxes. I think the cloak belonged to Godric. ...OMG...the Sword never became a horcrux and in fact was imbued with basilisk venom which destroys horcruxes. So if you're afraid of horcruxess wouldn't you want to keep a steady supply of basilisk venom on hand? Just saying... That's a MUCH better reason to run the risk of having a basilisk in a school than to get rid of muggleborns.
Well that's it. I supposed I shall refine this later...