The only way other people know that you are online Facebook at the moment is if they are your friend and you are on chat. You can become invisible on all your friends chat panels if you go to the "Go Offline" Option in your chat panel and click that.
Stop it.
Sneak it in.
No.
You can't. It's close to impossible.
It is unlikely to become a billionaire without anyone knowing, as accumulating that level of wealth usually involves significant business dealings, investments, or public recognition.
Invent mean that you make the object all by yourself without anyone knowing it, but discover means that someone already made that object but you just find without anyone knowing it.
Don't tell anyone
Make sure that all of your settings are set to private and that the chat is turned off and nobody will be able to know that you are on Facebook.
don't tell anyone
You can hire a private detective and he'll search for him for you
Exercise with nobody around
Make a facebook account and don't tell them that you have one. Don't add them as a friend either and if your parents have a facebook, don't send anybody who knows them really good a friend request.