Learn Russian: Он ни у кого не просил квасу.

This is the partitive genitive, sometimes called the "second genitive." For certain uncountable mass nouns like квас, чай, or сахар, adding -у or -ю indicates a quantity or "some" of that thing. So квасу specifically means "some kvass." You will also see the standard genitive кваса, but the -у form is very common when talking about portions of food and drink.

Why is the preposition placed in the middle of ни у кого?

When a negative pronoun like никто (nobody) is used with a preposition like у (from/by), Russian splits the negative pronoun. The negative particle ни comes first, then the preposition, and finally the declined pronoun. That is why it is ни у кого rather than "у никого". You will see this same sandwich pattern with other prepositions, such as ни с кем (with nobody).

Is квасу in the genitive case because the verb is negated?

It is actually working on two levels here! First, the verb просить (to ask for) naturally takes the genitive case when asking for "some" of an uncountable substance, which is why we use the partitive -у ending. Second, even if he were asking for a countable object, negated verbs in Russian frequently force their objects into the genitive case anyway.

Why do we need не before the verb if we already used (from nobody)?