"It's a Small World After All" is probably playing eternally in some layer of Tartarus after this, just as an added punishment Hades thought because of this.

Well, most versions I've seen is that the items of the helm, the lightning bolts, and the trident are not relics--they're were specifically made by the Cyclopes as thank you gifts to the gods for freeing them from being locked up. As for a fight, I actually can't recall any actual disagreement, it just jumps to them drawing lots because they're trying to decide which kingdom (heaven, sea, underworld) will go to whom.

... In case it isn't clear, I am a MASSIVE Greek Mythology nerd.