"You are not going home."
Translation:Nie idziecie do domu.
It would still be "do domu", as the Genitive form "domu" is here because of the preposition "do", not the verb. So no negation will affect it. Although negated Genitive = Genitive anyway...
The only sentence in which I can imagine putting "nie" in front of "do domu" would be "Idziesz nie do domu, [a/ale] do szkoły" (You're going not home, but to school). I really don't think it would work in your sentence, where you have the verb repeated in the second clause.
They are definitely don't interchangeable. The context of 'going to school' may make things confusing because it appears to be inconsistently used in English, but in general "iść" translates to "to be going" and "to be walking" (right now, or expressing a plan as in "Tomorrow I'm going to the cinema"), while "chodzić" translates to "to go" and "to walk" (with some regularity). It can also be "to be walking" if you're walking without any direction nor destination, just walking around.
On their own, "chodzić" and "iść" do not take any case because they're not transitive verb, they do not take an object. They need a preposition to denote going somewhere.