In one of Aang's visits to an airbender temple, a airbender game is shown with balls and poles. This, in addition to "Earth Rumble," suggests that sports involving bending were common even before "Pro Bending."
In Tales of Ba Sing Se, Iroh finds a handful of kids playing what appears to be an Earthbending-based version of Soccer. Since the URN began as Earth Kingdom territory, the game could have been adapted to include other benders.
Aang runs into trouble because his concept of 4 separate nations cannot accommodate mixed bender families, and he feels that harmony cannot be achieved otherwise, but what have he and Katara been doing all along? The Gaang too? Could the realization have given him the idea that the United Republic might work? He's already faced the issue at a personal level, now he shows the way in extending it to the rest of the world.
Aang is having trouble with current events because his understanding of the world is 100 years out of date. Part 1 and even the show itself highlights his antiquated concept of culture- especially Fire Nation culture. He's having difficulty understanding the situation because he's trying to pull Status Quo Is God with century-old 'quo.
Part of Zuko's struggle is that he realizes this, but can't find a way out of it.
Yu Dao is a hundred years old - the oldest colony, established by Fire Lord Sozin, perhaps the very one Roku broke up during "The Avatar And The Fire Lord", and the only one seen which has both Firebenders and Earthbenders, unlike Haru's village where they are imprisoned on identification, or Gaipan(the village Jet flooded) where there are no Earthbenders at all. What if "colonies" like Yu Dao are what Sozin intended from day one - a way of "sharing the Fire Nation's prosperity with the world?" The only problem was, Roku wouldn't even look at it! The United Republic of Nations might have come about a century and a half earlier if only Roku and Sozen talked like grown-ups instead of children who stole each other's toys.
Sozin not only turned out to be a Well-Intentioned Extremist, but it also turned out that The Extremist Was Right. Separation of the nations wasn't necessary for balance, and was in fact holding them back from the greater achievements seen in The Legend of Korra. The general idea is that there was good to be had from the idea if Roku had not been so dismissive of it out of hand. There was a way to do achieve the effect we see in The Legend of Korrawithout 100 years of war, ethnic cleansing, and all that. Roku could have steered Sozin towards those ends if he had put forth any effort in that regard. And if Sozin really did just want to conquer territory, then Roku could drop the hammer on him. And if Katara hadn't used her Cooldown Hug to point that out, Aang would have repeated Roku's mistake. Can we just say that Katara is awesome?
Of course Sokka got his boomerang back, but not his space sword- it's a boomerang. It's supposed to come back.
The scene with fire barrier of the rival Kunyo firebending school's students only warmed the metal coins Toph's metalbending students shoot at them. Thus said coins became dangerous to the firebenders than to the metalbenders. This explains why firearms seem conspicuously absent in The Legend of Korra even though the technology level could easily support it (Sokka had gunpowder in one of the original series episodes).
Aang thought he was faced with two ways to maintain world peace and balance: create a world where all elements live together or keep each of them apart. However, the only option that is feasible is the former, because Aang forgot that he is the LAST AIRBENDER! Should he decided to choose the latter option and followed it rigidly, he would leave no descendants behind, resulting in the extinction of airbenders that would permanently damage the balance of the world. In other words, while he might be able to maintain world balance during his reign, the balance would die with him. Thankfully, he made the right choice in the end.
Of course, the Avatar isn't immune to hypocrisy. That, or Aang has fallen into the trap Korra has: not really thinking out his future beyond being the Avatar. He's 13, after all, and how many 13-year-olds really think about their future marriages and children? Like Zuko, Aang is also a stubborn boy who isn't immune to acting rashly or without forethought. So he doesn't really think his whole plan through until Katara calls him on his hypocrisy.
Boomerang always comes back! ...So when did Boomerang come back this time? He's pretty certain it wasn't coming back when he and Toph were in dire straits on the Airship in the ATLA finale, and then... he suddenly has it again in Part One here.