"I do not eat soup."
Translation:Я не їм суп.
No, this sentence is in an accusative structure. I am doing something to something. Hence soup would be put into its accusative declension, however this particular word "soup" has the same nominative and accusative structures and so does not change.
With regards to the genitive part, this Ukrainian course only teaches genitive in the following:
у мене є and у мене немає. Any nouns after these must be put into their Genitive declension.
A negative only negates whatever its pointing to, in no way does make anything decline.
Can I say: "я не п'ю суп" as in "I do not DRINK soup" instead of "EATING" it? (just to know if it's okay to EAT a liquid in ukrainian)
It means "this".
I think there's a mistake and one of the multiple choice options is "I don't eat THIS soup", which is different from "I don't eat soup [in general]". I've reported it.