Other mistake: The Air People were wiped out by the Fire Nation through Suzon's Comet enhancing them, we even have confirmation of this by the dishevelled exposed skeleton of a VIP of them (presumably any Air Nomad who escaped would have given Monk Giatsu honorable burial), but we never see scorch marks on the Air Nomad buildings or the aforementioned skeleton.

Avatar: The Last Airbender (2005)
1 mistake in show generally - chronological order
Starring: Dee Bradley Baker, Mae Whitman, Jack De Sena, Zach Tyler
The Cave of Two Lovers - S2-E2
Iroh: Where are we going to go? We are enemies of the Earth Kingdom, and fugitives from the Fire Nation.
Zuko: If the Earth Kingdom discovers us, we'll be killed.
Iroh: But if the Fire Nation discovers us, we'll be turned over to Azula.
[Iroh and Zuko give each other a meaningful look.]
Zuko: Earth Kingdom it is.

New this weekTrivia: When Sokka gives Toph the piece of meteorite, she bends it into various shapes between her hands, the last of which resembles the Nickelodeon splat logo. (00:22:50)
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Chosen answer: Toph had Katara use it to send a letter to her parents. It never returned.