Я хочу торт :)

The verb is хотіти. Case is accusative, for this noun it's the same as nominative

Please type хотіти here: http://lcorp.ulif.org.ua/dictua/

You need to keep practising and get your ear more attuned; she definitely says не. It takes time! You'll get used to it.

If you say "I don't like cake" it means you don't like cake in general. "I don't like the cake" means you don't like some specific cake you have in mind. "I don't like a cake" doesn't sound very meaningful.

Now try to write this sentence in Ukrainian! :)

(I will help if something is wrong)

Nope, Duolingo is not sensitive to punctuation. You can skip any commas, periods or question marks you want in any language course (not only Ukr) and it will be accepted.

Please always provide your exact input. Probably you had a typo or some other mistake. It's usually displayed in the red text that pops up, pay attention to which words are underlined if any, and to the correct answer, compare it to yours.

Next time, provide the sentence you inputted in your comment :)