The Saxon genitive, while usually analysed as a clitic today, started out as an inflectional suffix for some noun classes, which was later extended, potentially, to all nouns/names, in the form <-es>, which was shortened to <'s> after the vowel was deleted in speech (everywhere but in front of sibilants). The pseudo-extended form "John, his dog" was actually a folk etymology, a (mistaken) reinterpretation of the apostrophe as standing for the shortening of a possessive instead of the clipping of a vowel.
Also, the possessive "its" is exactly what it seems: the (Saxon) genitive of "it". Well, "his" also is etymologically a genitive, but its derivation goes a lot further back in time (the ancestor of "her" was originally a dative, according to Wiktionary); the neuter possessive was the very same "his" in Old English while the newer "it(')s" completely replaced it for neuter duties only in the Seven-Hundreds. The problem with "its" is the fact that, although arguably it should, it doesn't align with the pattern of all other <'s> genitives by not having the apostrophe, which was reserved to the contraction of "it is": many spellers extend the pattern by analogy, in so committing one of the most frequent, and probably most venial, misspellings in English. (A misspelling I've found myself making a couple times recently - having learnt English through writing I'm usually subject to different mistakes than those native speakers make.)
Well, then, I stand corrected. Thank you.