Deker,

on negation: semantically, it's actually partitive (as in Хочу супа. Хочу чаю.)

Accusative expects a direct object, a known thing. It implies definiteness (mostly). "I know what I want".

Partitive (which is morphologically genitive in this case) expects, as its name suggests, a part of something. "A part of something" is similar to the notion of indefiniteness ("i don't how much" > "i don't know what").
This is how Russians came up with a way to express definiteness/indefinitess (apart from word order). In modern Russian, partitive is mostly merged with genitive.

Genitive: Я не вижу машины. I don't see any car. (However, rarely may be "I don't see the car" too!)
Accusative: Я не вижу машину. I don't see the car. (Only!)

In other cases, morphological genitive is used instead of etymological accusative in order to differentiate between subject and object (since word order is free). For example, without genitive we wouldn't know who is who in the phrase "Отец любит сын". "The father loves the son" or "The son loves the father"? Word order is free so both interpretations would be valid. Notice also that accusative->genitive shift works only for animate objects, because inanimate objects ("window", "house") are mostly passive as they can't perform any actions as humans do, thus they mostly stay in accusative, and there is no crucial need for them to differentiate between the two etymologically homonymous cases -- accusative and nominative.