No one knows the answer.
This is correct.
No one know the answer.
This is incorrect. "No one" is singular, just like "one" and "someone", so the plural verb "know" is incompatible.
There is nobody answering the question.
The grammar of this one is correct, but it has a different meaning.
This would be appropriate if you want to say no one is answering the question right now, e.g. if a teacher posed a question to the entire class but no students raised a hand to answer it.
But if you want to say no one knows the answer, you could phrase it as "There is nobody who can answer the question."
There is nobody answered the question.
There is nobody answer the question.
There is nobody answers the question.
These are all incorrect; you're missing the word "who" in the relative clause. "Nobody answering" is grammatically correct because it's short for "nobody who is answering" (just like "nobody famous"), but that shortcut only works with the -ing verb form.
"There is nobody who answer the question" is still incorrect, because "nobody" is singular and "answer" is the plural verb form.
The other two become grammatically correct once you add "who", but again they have a slightly different meaning from the other correct examples. "Nobody who answered" refers to the past; "nobody who answers" suggests repeated, habitual behavior.
For example, you could say, "Every year, I ask my new students when the Battle of Hastings was fought, and there's nobody who answers the question." (Although "...and nobody answers the question" is more idiomatic.)