"We do not see the coffee."
Translation:Nie widzimy kawy.
You somehow made a grammatically correct sentence with different meaning. In less "literal" meaning it would translate "we see wrong coffees" , I see coffees, but not the ones I need/want.
usually splitting "nie" and verb, either makes sentence ungrammatical or changes meaning signifficantly
Yes, it matters. "Nie" should be used before a word which it negates. So your sentence means something like: "We see NOT these coffees.". Moreover, there are common word orders, less common word orders and word orders which are simply wrong. The word order in a sentence can change depending what information you want to accent.
Is this a general rule ? Every time a verb is negated ("nie [conjugated verb in present tense]") the nouns following will be in Genetive ?