Thrasyl(l)us does seem to have had something to do with the transmission, but it’s far from clear what. Our main source both for Thrasyl(l)us and for the organization of the Platonic corpus is Diogenes Laertius (3.56ff.), where Thrasylus is said to have said that Plato published his dialogues in tetralogies, on the analogy of tragedians’ tetralogies. That’s obviously absurd. Originally the dialogues were put out individually, and that’s how they circulated for centuries, in roll form. It’s quite unclear how the tetralogic arrangement got established.
Thrasyllus is credited, rightly or wrongly, with the double titling of the dialogues—“Phaedo or On the Soul” for example. Socratic dialogue being effectively a new genre, dialogues could be labeled as if they were plays (Phaedo, Crito, etc, cf. Theseus, Medea, etc) or as if they were treatises (e.g. περι δικαιου, the alternative title of the Republic). I don’t know how Plato titled them, if at all.
They were variously classified (by neo-platonists?) as “ethical,” “logical,” “protreptic,” etc, the Republic being “political.” The organization of the Demosthenic corpus is comparable, and that controlled the transmission (but not the organization within any one group, curiously). But the grouping of the Platonic dialogues into sequenced quartets, which eventually prevailed, follows different principles (some more discernible than others), and its origin is obscure. It’s quite artificial (and requires the number of genuine dialogues to be a multiple of four). It can’t be very early.
Just as the individual dialogues vary enormously in size (from just a few columns to thousands, as if to defy book-making conventions), so do the tetralogies (the eighth comprises Clitopho, Republic, Timaeus, and Critias!), so it was not at all a practical arrangement, or not until the codex became sufficiently capacious in later antiquity, and hardly even then. It’s more of a theoretical organization.
There were other orderings too. Again Diog.Laert. is our source, and there’s no agreement on what his sources were. Apparently there was one arrangement in which the dialogues were sorted into trilogies (headed by Rep. Tim. Crit.). This one is associated (like so much else, including colometrization of lyrics) with Aristophanes of Byzantium, but Alexandrian organization of the corpus is unlikely to have been like that. DL records various alternative sequences. Philosopher-teachers’ opinions on the order in which the dialogues should be read were very varied, and this may have affected their sequence in some codex texts. There’s no evidence.
Thrasyllus does seem to have had a significant role in the transmission (reportedly he organized the Letters too), but his work will have been one of systemization. He won’t have exercised any influence on the text. There’s been much written about it but it’s all up in the air really.
As to –ηι vs. –ει, I neither know nor care. Ancient manuscripts would be expected to modernize orthography to some extent unless there was a strong tradition against it, but within any given tradition it’s rare to find consistency. I certainly wouldn’t use them to establish Plato’s own practice. In non-Attic authors the given orthography is often an imposition by ancient scholars and cannot be original, but that would hardly apply here, unless there was puristic insistence on the older spelling.