Hi Rodrigo
The words 'everybody', 'everyone', 'anybody', 'anyone', 'nobody', 'no-one', 'nothing' and 'everything' always use the singular verb in English. It's just a one of the English language rules - I know it doesn't seem very logical!
However, in your examples above you are making a negative statement, and so rather than use 'everybody/everyone' or 'anybody/anyone' in English we would always say 'Nobody does'. So, we use the negative word for the subject and keep the form of the verb as an affirmative one.
Ellie.