"- What do you want? - Nothing."
Translation:- Що ви хочете? - Нічого.
Both «Що ви хочете?» and «Чого ви хочете?» are possible here. The genitive generally adds a partitive meaning, as if you don't want the object as a whole but a part of it. E.g.:
- Що ви хочете? Чай чи каву? 'What do you want? [A] tea or [a] coffee?' (this is the default option)
- Чого ви хочете? Чаю чи кави? 'What do you want? Some tea or some coffee?' (this option assumes you don't offer all the thing, but just some part of it — so, it sounds OK for things that can be divided into parts)
If the thing cannot be divided into parts, and used as a whole, then accusative sounds better:
- Що ви хочете? Компʼютер чи планшет? 'What do you want? A computer or a tablet?' (here, we're talking about items as a whole, so we use accusative)
Since we don't have the context in Duolingo, I believe both options should be accepted.
The answer doesn't validate much here, because it's normal to use genitive for objects in negative sentences, and accusative in positive sentences. E.g. «Я хочу цю книжку» 'I want this book' vs. «Я не хочу цієї книжки» 'I don't want this book'.
The verb changes its form depending on the pronoun used:
- Що ти хо́чеш?
- Що ви хо́чете?
Mixing ти with хочете is ungrammatical. Although, of course, you will be understood if you confused them in real life.