Nobody knows the answer.
There is no one who knows the answer.
Nobody doesn't know the answer. (= Everybody knows the answer)
There isn't anyone who knows the answer.
Not anyone knows the answer.
Is there anybody who knows the answer?
Is there anyone who doesn't know the answer?
Does everybody know the answer?
Does anybody know the answer? (Is there someone who knows the answer?)
Neither of us could go there.
Either of us could go there.
Can either of us go there?
Which of us can't/can go there?
Can everybody go there?
Can anybody NOT go there? (Are there certain people who can't go there?) Can't anybody go there?
Is there anybody who can/cannot go there?
Not anybody can go there.
Not just anybody can go there. (Meaning only certain people can go there)
One of us could/couldn't go there.
Both (many, some) of us could/couldn't go there.
Both "either" and "neither" are incorrect in the negative such as in your example 2. Note that anybody/anyone are interchangeable and nobody/no one are also interchangeable. I hope this gives you a feel for how these sentences are formed.