Pattern of "why don't you, why you don't, ..."

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Some examples in the order you provided...
1)"Why you don't want to visit is beyond me."
2)"Why don't you want to visit this weekend?"
3)"Why do not you" --> "Why do you not want to visit?"
4)"Why you do not want to visit is beyond me."
5)"I don't know what happened with you, but why don't you visit for the weekend?"

(1) and (4) are the same. The first is simply a contraction of the fourth. No difference in meaning.

You are right that the nuances change based on the syntax (order of the words). The syntax for (1) is typically used when the speaker is talking to themselves about someone else, while the syntax for (2) is typically used when the speaker is talking directly to that someone. (Edit: Kieran's point is excellent...(1) is not a question while (2) indicates a question)

(3) is okay if you swap the words around as I mentioned above. In this case, (2) is simply the contraction of (3) and there are no differences in meaning. However, (2) sounds a bit better than (3).

(5) is also directed to a listener and thus has a similar meaning to (2) and (3).