TEAM FRED!
I think Fred is a manifestation of a specific problem experienced by Elizabeth. And yes, obviously, she is doing the things Fred is doing or at least imagined doing.
The scene in the psychologist's office: I saw that scene as a representation of "the crazy in me recognizes the crazy in you." Perhaps the children imagining what it would be like to just cut loose with all the other kids.
And at the end, Fred was with another child who was having the same problems Elizabeth was experiencing.