Flaying Foxes and Vomiting People

This isn't really an answer either—just another piece of information that is too long to fit in a comment. One of the earliest English slang dictionaries I'm aware of is A New Canting Dictionary: Comprehending All the Terms, Antient and Modern Used in the Several Tribes of Gypies, Beggars, Shoplifters, Highwaymen, Foot-Pads, and all other Clans of Cheats and Villains (1725). It has these entries for fox and fox'd:

FOX, a sharp cunning Fellow. Also Drunk ; as, He has caught a Fox, he is very Drunk.

FOX'D, Drunk.

If a person who has "caught a fox" (in the English slang of 1725) is very drunk, might it not follow that a person who is vomiting the residue of his drink is "flaying" that same fox?

Several searches of Google Books for various forms of "flay a fox" turned up nothing beyond the Rabelais translation that Silenus mentions and a handful of dictionaries that cite the phrase in connection with its slang meaning.

As long as I'm here, I might as well add that I don't find very believable the theory that "to flay a fox" means "to vomit" simply because flaying a fox is such a revolting operation. We are, after all, talking about an era when people set dogs on chained-up bears or bulls for entertainment. The Wikipedia article on bear baiting cites an instance in which "a pony with an ape tied to its back was baited." According to a spectator at that event "with the screaming of the ape, beholding the curs hanging from the ears and neck of the pony, is very laughable." I can't imagine that flaying a dead fox would be more nauseating than attending one of these popular amusements that involved tormenting live animals until they died.

Apparently, in the so-called "dark ages", women vomited foxes. Well... it looks like a fox to me, or a rabbit.

Unfortunately, I couldn't find the source of this particular image, which is an example of medieval marginalia.

Generally speaking, marginalia simply means anything written or drawn into the margins of a book. In the medieval context, marginalia is understood to mean images that exist outside or on the edge of a page’s main program. Collectors Weekly

Unlike the fox I lack his cunning, and I was curious to understand why the cant term, fox, should mean “to be drunk”; Charles Richardson in his A New English Dictionary of the English Language (1836) shed some much needed light.

FOX, v. To deceive, to entrap, to ensnare; and thus, to intoxicate, to make drunk

It also included the following citation from The works of Francis Beaumont and John Fletcher
Beaumont (1584-1616) Fletcher (1579-1625)

Clow. So so: as long as we kept the Mop-headed butter-boxes sober; marry when they were drunk, then they grew buzards : You should have them reel their heads together, and deliberate; Your Dutchman, indeed when he's foxt, is like a Fox, For when he's sunk in drink, quite earth to a Man's thinking, 'Tis full Exchange time with him, then he's sublest; but your Switzer, 'twas nothing to cheat him.

The Fair Maid of the Inn

Q3 Consequently, the final perplexing quote from Rabelais means the following

Pantagruael, the protagonist, meets a young scholar and casually enquires where he is from. The reply spewed by the scholar is nonsensical and infuriates Pantagruel who cries, “What devilish language is this? By the Lord, I think thou art some kind of heretick”. As the conversation progresses, it becomes clear among those who assist the encounter, that the young scholar is feigning to be a Parisian, he is only a Limousin “... because he disdaineth the common manner of speaking.”

Thus when Pantagruel cries: “Thou flayest the Latin” he means the pretentious youth with his highfalutin speech is slaughtering the Latin language, (maybe the upper class in Paris spoke in Latin) “... by St. John I will make thee flay the fox” he swears he will make him spit out (spew = vomit) the truth, “for I will now flay thee alive” because he will beat him alive.