If I recall correctly, the Iron Cage incident happened pretty much immediately after the formation of the Black Templars and other Successors, right? And that first break-up (the one before the Iron Cage) occurred as a result of those impending hostilities with the Imperium at large, yes?
No and no; the Iron Cage incident occurred about the time of the implementation of the codex astartes, not after it. At the time, Dorn felt the Imperial Fists to large to willingly break up into the chapters. Like Dorn, the fists themselves had taken the wounds of the Emperor to be a personal failing, though not to his degree, and until they had redeemed themselves, breaking up would be impossible. They had to be redeemed as Imperial Fists, nothing less would be acceptable.
The Iron Cage was the chance for this, a fight that would most definitely lead to much of the legion being wiped out as they took with them the vengeance so dearly needed. That Dorn underestimated Perturabo and the Iron Warrios like he did, thats what led to it becoming what it is known for. (Though always remember, despite everything Dorn and the Fists fought their way through overwhelming numbers and situations they were not equipped to deal with.)
It is very likely that Dorn had been so blinded that he fully intended to wipe out his entire legion, and the actions of Perturabo, and then Guilliman, are the only things that prevented this.
Why the Fists broke up is a combination of things. As mentioned, they needed redemption before it was to happen. But what got them to actually submit to the action of breaking up the legion would be when Guilliman played his hand and showed that he was fully ready to plunge the shattered Imperium into a second civil war to ensure that no one leader/warlord/tyrant was ever able to command so many with such influence.
Say what you will of that action, personally I think Guilliman did the right thing and in a noble way.