The Monstrous Strength explanation is an attempt to reconcile the human (and somewhat prominent) Caw found in Culhwch ac Olwen with the monstrous giant seen in a biography of Saint Cadog. Obviously, there are timeline issues, since he slays the boar on Arthur's horse and the giant at the behest of Culhwch, Arthur's younger cousin, but sod 'em. We'll say that after his and Hueil's death, the bards of the court appropriated Caw's deeds for Arthur's glory with the blessings of his family (though Hueil is also mentioned in Culwch ac Olwen, and he still retains his title of 'never submitted to a lord' and is mentioned as an enemy of Arthur).
For a possible etymology and interpretation of the 'thaleithyawc' title, I follow Rachel Bromwich in her notes to the Triads. For that matter, most of this is syncretised and culled from her book, so I owe a great deal to her indeed. The whole Noble Phantasm title means 'The Second Torqued One', where what seems to be an early version of Tristan is the first, and Kay the third.
Bai'r Glun is taken from the tale of Hueil catching Arthur in women's clothing, dancing with other girls. Having already wounded the king in the knee and having been sworn to silence about it in exchange for a peace treaty with Arthur, he noticed the tell-tale limp. He couldn't restrain himself from saying "Da iawn yw downshio velly oni bai'r glun" - "it'd be good dancing except for the knee". Having broken his promise, he was beheaded by Arthur.
My version of Hueil catching Arthur in women's clothing did turn out rather more reminiscent of a Shakespeare comedy (lolcrossdressing, lolmistakenidentity, loldisguises, lolfallinginlovewithpeopleyoushouldn't). Because, really, that's the perfect set-up.
The part about Gildas is mostly taken from known tales, probably made up to explain the lack of references to Arthur in Gildas, who would have been within a generation or so of him if he were a historical figure. However, the translations from Gildas' Fall and Conquest and their use in establishing a link with Arthur himself are mine. (The translation is accurate, though I've definitely taken the interpretations which help along the scenario I constructed).