Hiding email address

I have lots of users with fake mail addresser because they havent any real email.

I have created a condition with CB Conditional to hide those fake emails in Profile and User Lists, and my question is,

Is it possible to hide the email in the Message menu item in the profile too, with an override or whatever?

You could use CB Auto Actions and its Code action to dynamically add CSS to the profile page based off a field value in order to hide it. There's no other way to hide that menu item except using CSS. You'll need to act on onAfterUserProfileDisplay trigger.