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Aang is having trouble with current events because his understanding of the world is 100 years out of date.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 Sozin 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?
Yeah, except Sozin wasn't advocating peaceful trade or cultural exchange; he was foisting the Fire Nation's prosperity onto the other nations whether they wanted it or not as a transparent excuse to grab power, and the whole shtick about "sharing" was just a flimsy cover to get Roku to go along with it on the strength of their friendship. The "brilliance" would be in the fact that Sozin's flimsy excuse was what actually ended up coming out of it once peace had been achieved.
Of course Sokka got his boomerang back, but not his space sword- it's a boomerang. It's supposed to come back.
So why might Zuko have gone to former Evil OverlordOzai for advice on running the country, rather than the only remaining member of his family who actually cares about him? Well, Zuko did go to Iroh for advice when he was the one in prison, back during the TV series, and his uncle refused to talk to him. Ozai, on the other hand, offered advice without even being asked—albeit at a cost. Maybe Zuko figured that it was Ozai or nothing, and he might as well take his chances.
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).
Remember when Zuko passed out and slept for four days? Sure, he's catching up on lost sleep, but he's probably also recovering from that dislocated shoulder. We last saw him falling off of a cliff. Assuming that the Avatar planet has around the same mass as Earth, he's accelerating at a rate of 32 feet per second squared. So every second he falls, he's falling 32 feet per second faster than the second before. As he's falling, he has time to say "Aang, I don't—", which means that he's been falling for about a second and a half already. Moreover, he's falling face-first, meaning that when Aang catches him by the wrist, the rest of his body continues to drop, while his arm remains fixed. Estimating that Zuko is around six feet tall, and weighs 178 pounds, the resulting momentum means that it's surprising Aang is even able to maintain his grasp on Zuko's wrist in the first place, and it's definitely enough to yank an arm out of its socket.
Had it not for Grey and Gray Morality and realization of their own flaws, the Team Avatar is this close to becoming Designated Hero as they are basically deporting Fire Nation colonists without any clue that they are actually rooting them off their homes based on his obviousness over why the residents of older colonies are upset over being "returned" to their homeland compared to the newer ones. This show how much resentment against Fire Nation and restoration of pre-Hundred Year War status quo have entrenched among the people of Avatar-verse.
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.
It could just as easily be a new boomerang—it's not like Boomerang was a one-of-a-kind weapon, unlike Sokka's sword; replacing it can't have been too hard.
Another alternative is that Sokka simply got up and looked for it. Given his impressive calculative ability he could have figured out its approximate location and started looking for it there.