So Ukrainian also uses accusative for object of a negated sentence, like Russian and unlike Polish? Would genitive always be incorrect?
Genitive is usually used when you speak about something in general (all fruit, all people), and accusative when you speak about particular person, particular sausage that is front of you on the table. But it seems to me that this course doesn't follow that..
So can I take them as rules?
singular general - genitive
singular definite - accusative
plural - accusative
Regarding the English Translation: Shouldn't it be rather "I do not eat sausageS" or as StrapsOpinion suggested: "I do not eat THIS sausage"
In English it is correct to say "I don't eat sausage" when meaning sausage in general. One would also say "I don't eat pizza" meaning pizza in general. One wouldn't say "I don't eat pizzas."
If one does not eat a specific kind of sausage, then one would say "I don't eat this (kind of/particular) sausage" meaning that one sausage in particular and is most correct when including "kind of" or "particular."
In referring to a singular sausage on one's plate that one does not eat, for example, due to dislike of that exact sausage, one would say "I won't eat this sausage" or "I don't like this sausage" or "I don't want to eat this sausage" which is an entirely different sentence in both languages from "I don't eat sausage."
In English, 'sausage' isn't usually used with the indefinite article. "Do you like sausage?" "No, I don't eat sausage." "Please pass the sausage." You could say, however, "May I have a sausage?" if, for example, someone is passing around a platter of them, and you want one.
To say "I'm not eating this sausage." is the correct translation "Я не їм цю ковбасу?"
Yes, but the sentence for translation is "Я не їм ковбасу?" Without "цю." Furthermore, the short forms like "I'm" are often not accepted and should be avoided.