The Southern Air Temple - S1-E3
Other mistake: When the gang is in the Hall of Statues of Past Avatars, Katara points out the avatar cycle pattern of statues and Aang zeroes in on Roku's statue. The earthbender avatar statue next to Roku is of an unknown man with a sword, not Kiyoshi. (00:12:50 - 00:13:35)
The Southern Air Temple - S1-E3
Continuity mistake: When Gyatso is laughing in Aang's flashback, he is missing quite a few teeth. But when Aang finds his skeleton, all his teeth are still in place.
Chosen answer: Toph had Katara use it to send a letter to her parents. It never returned.