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?
This is quite a nice answer to your question.
In response to your first question, here's what I would say:
How do you know? - How does the person know the information (present)
How did you know? - How did the person know the information (past)
The above examples mean the same thing, just one is in the present and one is asking about the past.
How would you know? - Probably used in a response/remark, e.g:
A. I think you stole the money
B. How would you know?
You can see in answer B that you can put the stress on would or you, to differ the meaning. E.g:
How would you know? - Asking how the person would know the information
How would you know? - Asking how the person in particular would know
Overall, the difference between do and would in this context is:
How do you know? - this is used when wondering how someone found out: e.g:
A. I know you stole the money B. How do you know?
As if to be admitting to it - the secret is out
and
How would you know? - probably a remark, the person thinks that the other person will not know
A. I know what you were talking about. B. How would you know?
This could be true or false, the person is not replying directly, but is just stating how the person would have found out.
In response to your second question, again, you should look here.
Best Answer
Let's examine Will we go? first:
You are in a group of people, speaking for the group, and asking the entire group if everyone in it wants to go.
It can also be a shortened form of Will we go next? - in a situation such as a group waiting their turn for an amusement park ride or similar.
This is will we go in the negative (will we not go?) You are in a group of people, speaking for the group, and asking the entire group if everyone in it does not wish to go.
Sometimes questions use this negative form (auxillary verb + not) as a shallow form of indirection, "softening," or politeness. The positive form is more clear but also more direct which may not be wanted in somewhat more polite or "begging" situations:
"Won't we go?" follows the same principle.
Similarly:
You are asking someone if you are going to leave.
Typically, the stress will be on the I, meaning the speaker/writer wants to confirm he/she is the one that is going to leave.
This also may come up if the speaker/writer is discussing a plan of some sort and wants to know if he/she will leave at some determined time, the determined time being understood from the context.
As explained with won't we go, this would be an indirect, "soft", or polite form of will I leave.