Thud!/WMG

The Battle of Koom Valley is a fractured Parrot
Warning: Paranoia Fuel; It's even harder to keep from hearing a sound pattern than seeing a wall. One of the sounds stored in Methodia Rascal's Device was an auditory version of The Parrot from Blit. He started "hearing" the Chicken, hearing it get closer... and closer... as it honed in on his position in the realm of his mind, he becomes less and less attached to where his mind is than the Chicken does, causing increasing "errors" in not only his reality, but his painting as well. He heard the fractured Parrot/Chicken from the Device, and drew it into The Battle of Koom Valley. When it was complete, he had painted the last bit of the Chicken, and it all fell into place (causing his mind to fall out of place). The feathers were just a result of his belief in the Chicken convincing the universe too that it existed. Alternately, it could be an auditory version of the Summoning Dark, sealing the -ness, which then escaped in the actual form of a Mind-chicken. Alternately alternately, at least two Darks could be Parrot patterns, and Vimes was just Knurd enough not to be affected beyond repair.
 * Being knurd, it seems, would just make it worse. The reason Vimes is fine you're looking for is pure, unadulterated, badass.

Rascal was really a chicken
To be more specific, a chicken soul that somehow ended up being reincarnated as a human. He was crazy because he subconsciously knew he was in the wrong body. The Chicken chasing after him was really Death of the Chickens, trying to kill his human body so he could be reincarnated properly.
 * Slight problem - Rascal was running from the Chicken for more than the mere amount of time covered in Reaper Man, the only period when a unique Death of Chickens could have existed. Death was alone before then, and the only survivors of his reassimilating the other, specific deaths into himself were the Death of Rats and the Death of Fleas.
 * He was a chicken trapped in a man's body. He was afraid of his inner chicken, because most people saw this kind of thing as some sort of abomination. It was all very metaphorical.

Rascal became obsessed with chickens because a Fury attacked him at Koom Valley.
The Furies (seen in Unseen Academicals) resemble chickens, and as flying creatures they're among the few races that could've safely settled Koom Valley for any length of time. Birds tend to be fascinated by shiny objects, so Furies living in the Valley could have noticed the Cube centuries ago, and considered it theirs, although they lacked the means to remove it from its chains. When Rascal stumbled upon and removed it, one of the Furies spotted the "theft", and tried but failed to wrest it away from him; having never heard of Furies, he named her "the Chicken" in his own mind. When he returned to Ankh-Morpork, the Fury stubbornly came after him, berating him from the shadows with chicken-squawk accusations, just as the Sisters harassed Nutt.
 * That would be why the cube started talking in the first place, Rascal didn't start it, it was the Furies' "Awk! Awk!"

The Watchman in Vimes' mind in Thud! is from Watchmen
It's actually Rorschach. Dr. Manhattan sent him there to protect Vimes, and so that he could truly become the hero he always wanted to be(or something like that).
 * Rorschach? Oh, come on; Vimes is pretty much ruled by the letter of the law. Rorschach is willing to be a vigilante, which effectively makes him a special kind of criminal (i.e., he operates outside of the law, and given the way Rorschach operates, he usually commits crimes like assault in the course of his work). You could argue for Rorschach having a similar tendency to snap and attack his "perpetrators" to Vimes', but if so Rorschach's is not only on anywhere near as tight a leash as Vimes', but it's probably fairly completely uncontrolled. Vimes is way more badass than Rorschach.
 * Yeah. if anything in Vimes' mind is Rorschach, it's the Beast that the Watchman guards against.

The Devices were made by Trolls.
Trolls get smarter as it gets colder—Detritus almost solved the universe when he got locked into the meat-futures freezer. The Devices are the last surviving traces of a Trollish civilization that flourished the last time the Disc went through an ice age. In Thud, it's mentioned that Devices are usually found under mountains, and in Sourcery, a long-ago ice age is mentioned.
 * Or perhaps it fell due to the fifth elephant. Vimes reflected that the trolls could have been there when it hit.

The Cube from Thud! was once baked inside the original Scone of Stone.
In The Fifth Elephant, we learn from the "Ironhammer & Bloodaxe" opera that a piece of "truth" had been concealed inside the original Scone, so that it could be smuggled past these dwarf heroes' enemies. The Scone came into the possession of B'hrain Bloodaxe, who died in the caves under Koom Valley, and had a Cube on hand to record his and the troll king's testament to the world. Why else would B'hrain have been carrying such a priceless Device with him, or personally owned one at all, if not because he'd broken open the first Scone in secret and extracted the Cube—a Device recording a great truth, namely a recitation of the unaltered Things Tak Wrote—so he could hear Ironhammer's voice again, after his beloved's suicide? He'd secretly re-baked the Scone from its shattered gravel, the first in a long series of dwarfs to do so, and kept the Cube that preserved Ironhammer's recital, adding his own to them when his death was imminent.

The Summoning Dark is somehow related to hivers.
It gets into people's heads, and makes them act bad-tempered and subtly out of character. A sufficiently strong will can stop its influence, or even drive it out completely (as seen when the SD meets Vimes's inner watchman, or when the hiver is tricked into the "soul and center" of Tiffany's mind). The Summoning Dark itself remarks that it had spent millenia just drifting aimlessly across the multiverse before "spending the last 10,000 years as a superstition".

The Summoning Dark is The Darkness.
Incredibly ancient quasidemonic thing that needs a host in order to manifest and feeds off of their negative emotions like anger and hatred? I can't be the only person who read its lines with Mike Patton's voice in my head.

The Inner Watchman is actually DW's god of justice.
Since we know how discworld, and Cori Celesti function, it is a safe bet that if there was ever a god of justice, and it is just one of those gods you just have to have, I mean even if no one actually would pray to him/it, there is a little piece in any person that believes in part at the actual concept of justice. So, we have a relatively minor god, that would never be welcome in the big place anyway because everyone there cheats, with nowhere to actually 'live as it is' except pure ethereal plane. On the other hand, we have Sam Vimes, who self admitedly does not trust in the gods, but he does fervently, and with all of his formidable willpower believe in justice, because without it he would be The beast. And, we know from Small Gods that one sole adherent of sufficiently strong belief can make a god, and also that the nature of the adherent in that case would affect the nature of the god in question.
 * Or alternatively, Vimes is going to become the God of Justice/Watchmen after he dies, in much the same way that  in Monstrous Regiment. We know it's possible now, and Vimes' reputation and the way the rest of the watch look up to him is reminiscent of.