This is well-written and very interesting. Probably the best Harry-as-Horcrux explanation I've read.
Unfortunately that theory fails in the face of Dumbledore's "As we know, he failed." Voldie did not make a Horcrux that night in Godric Hollow.
It is not enough that 'someone must die,' they must be murdered by the person making the Horcrux (as Book 6 says must happen). If Myrtle's death is how Tom made the first Horcrux (as seems likely), then the Basilisk is simply the means (not the agent, as 'at the hands ... of' implies) of the murder -- as a knife, gun, poison, etc. are the means of murdering.
Would Voldie refer to Harry as 'Dumbledore's prophesied hope'? Voldie did not know Harry was the one (Neville, after all, was just as possible). After he killed Harry, he was surely off to kill Neville. It's just that his plans were interrupted, so to speak.
Biblically, it is the soul that is the totality of the person, not the spirit (e.g., Gen. 2:7 God breathed the breath of life and man became a 'living soul').
You speak of the soul pieces as sevenths (three-sevenths, one-seventh, etc.). While convenient, is it accurate? Can you see Voldie with a ruler measuring his soul each time he makes a Horcrux to be sure it is a precise measure? He'd already made one before he spoke to Sluggie about more -- how did he know to make that first one a seventh? Or was that first one half, the second a fourth, and so one so that Voldie has only one-sixty-fourth of his soul left in him? Or does he have the majority in himself with only little pieces in each of the Horcruxes? Or maybe the amount of soul torn depends on how heinous the murder to perform it is, so that each Horcrux has a different sized fragment?
Why would the soul fragments go to Harry? It is understandable that the one from the Diary might -- Harry is right next to it when he destroys it (does this fit the depiction in canon where Riddle disappears screeching? It seems that is the soul piece being reanimated by Ginny's life force and vanishing when Harry destroys the Horcrux. Where is the indication that the 'blurry, through a misted window' Riddle moves toward Harry's scar? He is just gone after screaming and writhing). But why does the fragment from the ring come to Harry? Is he closer (Hogwarts is in Scotland somewhere and Little Whinging is near London, far to the south)? Why doesn't that fragment enter Dumbledore? Why don't the pieces from the locket and dagger go back to Voldemort? Wouldn't the original 'spirit' or 'body' would be preferred?
Sorry to ramble on so. This is a great read and thought provoking.