No, you can't use genitive case.
Genitive case is used for direct objects in two situations:
- in negative sentences (but not in all cases; e.g. you never use genitive when talking about people), like your example «Извини́, я не зна́ю отве́та»; accusative often works in negative sentences too (so, «Извини́, я не зна́ю отве́т» is correct, too);
- when meaning 'some', taking a part of the whole: «Я пое́ла ри́са» 'I ate some rice' (=not all of it, but 'some rice'); this doesn't work with 'answer' since you can't usually divide answers into parts.
Because this is not a proper sentence in standard English? "Is knows" is not an acceptable verb combination.
- If you have "is" in the positive sentence (She is OK, he is playing) then you move it to the beginning of the sentence to form a question (Is she OK? Is he playing?). The same is true for the other forms of the verb 'to be'.
- If you don't have 'to be', you normally use the auxillary verb "do" and the bare infinitive for the main verb (He knows the answer. ⇒ Does he know the answer?),
- although with some verb forms you can move the verb form to the beginning (He has been playing. ⇒ Has he been playing?).
So, since "He is knows the answer" is not a correct English sentence, neither is the question "Is he knows the answer?".
(Disclamer: I'm not a native speaker. But I believe my English is good enough to answer this question.)
Зна́ешь (znaesh) is the second-person singular verb form, it's used with the pronoun ты (ty) 'you'.
Зна́ет (znaet) is the third-person singular verb form, it's used with the pronouns он (on) 'he', она́ (ona) 'she', оно́ (ono) 'it' and with any singular noun.
It's similar to the difference between 'know' in 'you know' and 'knows' in 'she knows' (although Russian has 6 forms like this, while English only has two).