Rather then see these collection of behaviors as indicative of a personality disorder it is probably safer and easier to say that they make you feel uncomfortable. Nothing more after this needs to be identified.
What I hear very clearly from you is that you do not want to be friends with her. That is the bottom line. I would not invest energy or time in the relationship and then keep a close watch just be sure that your daughter is treated well by her. If not you may want to follow up by talking to the school’s principal.
Everything about her approach and interaction with you seems to have raised many red flags. Don’t ignore them. Worry less about whether or not she has a disorder, but find ways to be consistent with your behaviors and interactions that inform her you are not interested in a friendship.
Diagnosing her is less important than understanding how you feel about her.