Those are good questions, I've always wondered them myself.
I think alot of beeping her name out might have been an element used in the old Kung Fu movies. Just one of the many things to make it feel like one. Although when she is finally introduced as Beatrix Kiddo, I think the reason it goes into the class room scene is just to reinforce the fact that we are now being introduced to character, even if we have already seen her.
Very bleek, but always my take on it.