Yes, she's immortal. What part of that hasn't gotten through at this point, after she's talked as if she's been in Nara over a thousand years ago, and the chapter splash images depict her traveling the world in different time periods and countries? She's even talked about visiting Paris in the Q&A sessions, though we don't know exactly when that happened. It was likely a good while ago, at least, though. Various answers she's given in those have pointed very heavily at the fact she has been around for a very, very long time, and expects to continue being around for a very, very long time.
However, she is not 'returning to the moon'. She didn't come from the moon. She is, by her own admission, a normal girl. She was born in Japan, but over a thousand years ago. But best theory I can figure is that she somehow received the immortality elixir, from the Tale of the Bamboo Cutter when it was meant to be burned and destroyed, during her 16th year of life. This is the connection of Kaguya-hime's tale to her, and nothing further, based on my own understanding.