Nothing grammatically. Northern Chinese would tend to say 那儿 and Southerners 那里

I'm just a beginner, but I think 那儿 is Beijing dialect. In other parts of China, they say 那里. As far as I'm aware, they're interchangeable.

One more used in the North, the other more used in the South.

There's no difference, except 「哪里哪里」 can be used as a polite response to a compliment.

There isn't any real difference between them.

Well, she's also not there should be correct. 在 doesn't always mean inside

Actually, you cannot say "neither not".

Both "She isn't here either" and "Neither is she here" should be acceptable, but "She isn't here neither" is definitely non-standard.

Personally, I would see that double negative as additional emphasis that she isn't in a particular location, like "You say that she's here, but I'm telling you she isn't!"

I hear it as a third tone but a very soft one.