Did Anne and Peter fall in love?

Yes they did, though in Anne's case, it was more a strong infatuation. Once Anne realized she had Peter's unwavering devotion, she somewhat "lost interest", and began to see his worst faults more clearly. It is hard to say what Peter's true feelings were, though he probably was in love with her, at least briefly. But Peter became very emotionally "needy" of Anne, and Anne would have none of that.