The problem with this is that unlike the runs or scissors or the heebie-jeebies or any other example I can think of, The Splits has multiple forms of use that necessitate a singular form. No one is ever concerned about having "a run" in regard to making it to the toilet. The Splits starts out sounding wrong but then quickly devolves into being un-useable when you have to describe a particular ...
What is the meaning of the following sentence? You have successfully split a hair that did not need to be split. Source: this post on the Programmers Stack Exchange.
For the most part, the words are interchangeable. Distinguishing between multiple examples of such things can be aided by their individual connotations: crack a line on the surface of something along which it has split without breaking into separate parts A crack tends to be a visible flaw that can splinter or spider into larger cracks with many smaller, attached cracks. The defining point of ...
Does the "in" imply multiplication, in which case split in half is correct, or is it division? It sounds like the latter to me, but I've heard it used both ways.
Every entry has a word split into syllables, and technically speaking, according to traditional rules of typesetting, you can hyphenate a word at any syllable boundary. For example in the Merriam-Webster's online dictionary, the entry for "dictionary" reads "dic·tio·nary"—so you could hyphenate anywhere there appears a centered dot.
Split infinitives involve the to-infinitive specifically. The "to" not a "preposition"; it is a infinitive marker. Lastly, I found your arguments about "wanna" & "gonna" unconvincing and irrelevant because these words are informal and the argument about split infinitives is most certainly about prescriptivism.