Ayu's Ark Had Me Completely Lost

That is correct. For most of the story the Ayu everyone is interacting with, the one wearing her duffle coat and winged backpack, is actually a type of spirit called an Ikiryou in Japanese folklore. This is basically a "living ghost", the spirit of a still-living person that has come out of their body and moves around freely in the world. Yuichi and his friends were meeting and interacting with Ayu's Ikiryou while her body remained at the hospital in its comatose state. To all appearances an Ikiryou is solid, real, and fully human, but according to the legends it will vanish if its true nature is ever discovered (which is essentially what happened - Ayu eventually realized herself what she was, which triggered her own disappearance). The Ayu in the last scene, sitting in the wheelchair and trying to hide her embarrassing haircut, is the real Ayu. All of her non-flashback appearances prior to the last episode were just her Ikiryou form.