I had the same question in my mind, but i could solve it. Direct object is accusative case, and for inanimate masculine nouns, accusative is the same as nominative. And it is not genotive because it is not after нет, and the negative is for the verb, not for the noun also :)
As far as I can tell there are no genetives in this sentence. I think there are two reasons for this.
1). Сок is a vocabulary word we learned in this course, so this is basically just vocabulary practice.
2). A sentence that doesn't have genetives is presented to contrast with the ones that do have genetives. This helps show when you do and do not use genetives.
They are the different conjugations of the verb пить (to drink). The verb changes depending on the person:
Я пью (I drink) Ты пьёшь (you drink) Он/Она/Оно пьёт (he/she/it drinks) Мы пьём (we drink) Вы пьёте (you all drink) Они пьют (they drink)
The "ё" is usually written as "е" in everyday life. I hope this helps!
Strange that "My sister isn't drinking juice. is marked as incorrect.