Yes, that's how it works. You use genitive when it comes to negation. Widzimy kawę. (Accusative) --> Nie widzimy kawy. (Genitive).
Yes but when it comes to the object of the sentence. You still use all the corresponding cases with words that are not the object of the sentence even if it is a negative sentence.
Nie idę chodnikiem. - I am not walking on the pavement. Chodnikiem is the instrumental of chodnik (pavement).
So in this sentence there is no preposition, the verb "iść" is simply structured to always take the instrumental whether negative or not? Or am I missing the point? :P
You are basically right, we have "iść + complement of place in instrumental" which means to go/walk somewhere. So we have e.g. iść ulicą, chodnikiem, drogą (which mean quite the same as iść po ulicy, po chodniku, po drodze (we use the locative case after "po" when it means on or over referring to places)). But back to iść + instrumental. We always use the instrumental here, nevertheless the sentence is negative because the word in instrumental is the complement, not the object of the verb. @DOWN: Thanks, no problem, anytime! ;-)
If the genitive is used for negations, is it possible to skip the particle nie the same way you can skip some pronouns? Like:
Widzimy kawy = We do not see the coffee
No, it's not possible to omit "nie". Here "Widzimy kawy" would mean "We see [the] coffees"; e.g. different brands of coffee on a shelf.
Nie widzimy kawę was accepted as correct (why? Or is it a bug/error?) that's why i came here and read it should be Nie widzimy kawy!