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My question is: if the world was breaking, lines of communication across the globe were totally broken, chaos everywhere, Aes Sedai fragmented and messed up so bad that they gave the Aiel all their stuff THEN went off and forgot half of what they could do with the Power, how does history remember that Lews Therin went mad and killed his family AND THEN killed himself, creating Dragonmount in the process?

There was no one else around to see any of it except Ishamael, and there's no way that a lone man, Forsaken or not, let everyone on Earth know to the point that the story survived the Breaking. I mean, in tGH, Rand fights the "DO" in the sky for thousands to see, and still half the world denies it. So one man's word definitely can't be enough, especially the word of a Forsaken.

But fast forward three thousand years and even country bumpkin Rand, a guy who knows jack all about the world (didn't even know he lived on land that had once been a hallowed kingdom), is talking about Lews Therin Kinslayer going mad and killing his famiy before killing himself and making Dragonmount like security footage caught the whole thing.

TL;DR: Other Aes Sedai would know what happened after seeing the scene at his home.

Lews Therin Telamon was the standing Tamyrlin Seat and leader of all Aes Sedai. Yes, the Breaking was chaotic and lines of communication were down, yet that is all the more reason that the other Aes Sedai would be attempting to find Lews Therin. It's very likely that every channeler in the world (or nearly so) felt the creation of Dragonmount, and the only male channeler left of such power would be Lews Therin.

One of the first places they'd check is his home, and any Aes Sedai of the age would recognize use of the One Power. From there, it's an easy leap of deduction to figure out what happened.

The story of Lews Therin would become a warning, not only among the remaining female Aes Sedai who went on to found the White Tower, but also among the remaining male Aes Sedai... not all of whom went mad at the same time. From there, it would undoubtedly be used as a cautionary tale against male channelers and the damage they could cause.

Or, to put it another way... If you were the White Tower and wanted the populace to turn in any male channelers they find, how better to do it than spread the story of how the greatest channeler of all time popped his cork one day and killed all his kin?

First I would argue Lews Therin as Tamyrlin was basically the single most powerful guy in the society. People tended to follow his tweets, especially the powerful.

Second, even though the world was breaking, I'm pretty sure Aes Sedai calm was, as in Rand time, all the hype.

The initial panicky response would soon be overruled by todo lists and tickets assigned all with the aim of disaster recovery. Information gathering is pretty central so for sure, maybe some small time passed but soon enough, some "brown" like Aes Sedai would show up, possibly investigate the bodies and conclude they were murdered with the Power. This could possibly at a later date be paired with the Dragonmount suddenly appearing where it wasn't before.

I suggest a mountain-at-death-height-to-power-ratio document didn't exist but then again maybe others were documented to be later compared against. Possibly this being one of the biggest (high power => high mountain at death by burnout) accurate assumptions could be drawn.

Again, my view is that Aes Sedai calm/clarity of judgement permeated the chaos as that is and was basically their main job requirement.

I need to get home to my books to find the proper quotes but I would also put forward that Ishamael was partially responsible for this. It is said somewhere (either of the WoT Companion or one of the later books, I'll have to look it up) that Ishamael was bound more loosely than the other Forsaken and was able to, on a time, touch and influence the world. It would be in his best interest to keep the tale of the "Kinslaying" alive to promote disunion and make the Dragon's job more difficult when he was reborn and needed to bring the nations together to fight Tarmon Gaidon.