Doubts about the registers of "have got" and "no *noun*"

Dear every teacher, I would like to know two matters.

Firstly, I understand that "have got" means "have" and also "must". I, however, am not sure about its register. My British friend guessed that it is most likely a slang, informal type, and should not be applied in decent writings like school reports and essays. Therefore I would like to know about your view.

Secondly, I sometimes do hear that some actors in computer games and movies mention something like "I am no hero.", "I am no killer." I know these phrases virtually mean "I am not hero or killer". But I would also like to know their registers as well. These are the informal sorts and should only be used in very daily non-serious speech, am I correct?

Thanks much for everybody's helping hand.