Why is "I do not go to school today" wrong?
"I do not go" either means that you don't go to school at all, or you regularly 'don't go' there. But here it's Present Continous used in the 'future' meaning, because it's actually "I will not go to school today".
How would you say then: "I do not go to school today" in polish?
Well, to my understanding I wouldn't say that because it's wrong in English.
"I don't go to school today" is accepted, which is an abbreviation of "I do not go to school today" both are acceptable in English
is Nie idę "I'm not going" or "I won't go" or "I don't go" specifically, or does it work like all of these in different contexts?
"I'm not going".
It may be used in the future meaning, so as "I won't go", but technically that's "Nie pójdę".