So basically, "quoi" is accusative case, right?

Actually, just as with English what, the word quoi serves several functions.  In the sentences in this lesson, quoi is in fact a direct object pronoun, not an interrogative pronoun.  In the sentence "Il ne sait pas quoi faire," quoi is the direct object of sait.  And in Wladuszka's native Polish (and other Slavic languages), the direct object is rendered in the accusative case.  So Wladuszka is on the right track, even though English and Romance languages don't decline their nouns with grammatical case endings.