As the previous commenter said, nobody, anybody and anyone are written as one word. No-one and No one are acceptable, but don't write that as a single word *"noone" (wrong!) As for your second question, yes, they are written with singular 3rd person verbs. However, "anybody" and "anyone" are generally used in questions, therefore your example sentences with those words look a little strange. :Does anybody/anyone like Miley Cyrus? I don't!"

Hi. Nobody is actually one word the way you are using it here. Nobody answers the door. Anybody opens the door. No one is perfect. It is always one word. Anyone opens the door.

Anyone... Anybody... and nobody are pronouns. Any one = adjective and a pronoun. Same thing for no body and any body. For example... though could say... the detective found no body in the morgue. Or... Dr Frankenstein would accept any body for his experiment.