The Desert - S2-E11
Other mistake: At the start, when Toph says, "I can hardly feel any vibrations out here," Aang moves his head, and it briefly duplicates. Right after that, Aang seems to shift into another position without any animation, most noticeable with his head and face. (00:01:56)
Chosen answer: Toph had Katara use it to send a letter to her parents. It never returned.