As the others said, both "kennen" and "wissen" are possible in this sentence. I think the general explanation is that "wissen" means "to know" in the sense of "to know a fact", and "kennen" means "to know" in the sense of "to be familiar with someone/something". That's why you can't say "Ich weiß den Mann" or "Ich weiß London" - people and places are not facts. However, in the sentence in the exercise, both meanings are possible: you can know (wissen) an answer, i.e. a fact, and you can also be familiar with (kennen) an answer.
'nicht' should come last - Ich kenne die Antwort nicht.
The position of 'nicht' is tricky. It will always follow 1. the finite verb (Ich arbeite nicht.) 2. nouns and pronouns used as objects (Ich kenne die Antwort nicht.) 3. specific adverbs of time (Ich mache es heute nicht.)
It will precede: 1. predicate nouns and adjectives (Er ist nicht nett. Er ist nicht mein Freund.) 2. adverbs (Ich mache es nicht gern.) 3. general time adverbs (Wir gehen nicht oft.) 4. prepositional phrases (Wir gehen nicht ins Kino.) 5. elements in final position; infinitives, past particples, separable prefixes (Ich kann nicht helfen.)
It can also precede a word if you want to give special emphasis. (Ich mache es nicht heute, sondern morgen.)
That depends on the context of the question in regards to the conversation. It makes sense if they are discussing some previously answered question. In this case they aren't asking if they know and are to reveal an answer but rather to discuss a previously answered question.
It cannot, and does not have the same meaning. Your question "can you answer" asks if someone is able to answer (a question). This sentence, "kennst du die Antwort" has to do with whether or not one knows (kennen) the answer, not with the ability (können) to answer. Your sentence suggestion would be translated as "Kannst du (die Frage) beantworten?"
Both are the 'du (informal you)' forms of the verbs. 'Kannst' comes from the verb 'können (to be able to)'. 'Kennst' comes from the verb 'kennen (to know, be familiar with)'. As you hear more and more examples of similar words (and spoken German in general), your ear will more easily tune in to the vowel differences.