This entire usage is very formal and not used in everyday speech.
- Will you not go to school today?
In formal use, this would probably be the most likely.
- Will not you go to school today?
This sounds like it's straight out of Shakespeare or Biblical texts. It's an archaic form that isn't used much any more:
You have heard, see all this; and will not you declare it? Isiah 48:6 KJV
I'm sure I've heard it in Shakespeare, too but this is what came up first.
- Won't you go to school today.
This one is the least formal but it's an odd combination of formal (the will you not) and informal (the contraction).
If I were going to ask someone this, I'd say:
Aren't you [planning on] going to school today?
Are is present tense because planning on is implied in the statement.
Or, if I'm asking about tomorrow, I'd say:
Are you [planning on] going to school tomorrow?
Will you go to school tomorrow?