"Alexandra de Hanovre" is what she is called in French references because Hanovre is just the French spelling of Hanover, which is Hannover in its German (and, oddly enough, Spanish) spelling. Yes, Spanish media, I've noticed, always use the original German spelling with the two n's. But French media, and Monegasque, always spell it Hanovre. So that's why Alexandra is known that way in Monaco. I believe she probably spells it like this privately too, because she is Monegasque/French from her upbringing, culturally, even if she isn't actually Monegasque in a strict sense. She may be Austrian born, but she has grown up in a suburb of Paris, and now in Monaco, so really, she must identify mostly as a French/Monegasque.
Good for her!
Yes, I've always thought so myself especially around the eyes and the shape of the brows. Her face is fine-boned and you can see her developing maturity. I'm sure she'll be a beauty as well, given her parents' good looks. But she does seem to have a charming and outgoing personality.
