Other mistake: When Azula attacks The Western Air Temple, there is a close up shot of her when she says that she is about to become an only child. Pay attention to the smoke coming from the airship behind her, especially above her head: it doesn't move. Even if this was remotely possible, it does move when the camera zooms out, making it a mistake nonetheless.
The Ember Island Players - S3-E17
Other mistake: When they are watching the recap play Toph says she can't see it because they are on a balcony, but when her character comes on she describes him as buff.
Chosen answer: Toph had Katara use it to send a letter to her parents. It never returned.