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?
Forget "no" and "not" for a second - they seem related, but it is a false comparison.
Instead, think about
"There is" vs. "There is not". These are related expressions but one is positive and one is negative.
Things are simple when the (reflexive) subject of the sentence is expressed positively.
For example, suppose the subject is "life".
There is (life on Mars).
There is not (life on Mars).
The complication is that sometimes the subject is expressed negatively.
Suppose the subject was "no life".
The verb-positive sentence remains the same.
There is (no life on Mars).
The verb-negative sentence has to change, though, because English (unlike Spanish) has rules against double negatives.
It is incorrect to say:
There is not (no life on Mars).
Instead, we say
There is not (any life on Mars).
Other examples of a negative subject becoming positive because the verb is negative:
There is no reason to agree with him. There is not any reason to agree with him.
There is no way out of this situation. There is not a way out of this situation.
There is no one alive who remembers him. There is not anyone alive who remembers him.
In each case, the "not" in "There is not" is part of the verb, making the verb negative, whereas the "no" is part of the subject, making the subject negative. So you can't really compare "there is no" with "there is not" - just remember that in English, you are not allowed to have both the verb and subject expressed negatively.
Best Answer
This is a very simple example of subject–auxiliary inversion, and it is required in most interrogative sentences in English. The subject and the auxiliary verb appear in the reverse of the order in which they would appear in a declarative.
Consider the following declarative version of your interrogative:
In your interrogative version, the presence of the auxiliary verb do causes subject–auxiliary inversion to take place. In negated questions like this one, the negating word not should appear after the subject, not after the auxiliary:
Note that the negating word can also appear attached to the auxiliary in the form of the contraction don't: