1) "Party pooper" is a slang expression which means somebody who brings down the mood, somebody whose presence stops others from having fun. It is something you don't want to be. The joke is interpreting that expression literally.
2) Okay, I think this one is really funny! The phrase "write on an empty stomach" is ambiguous. To do something "on an empty stomach" means to do something while very hungry. But "to write on an empty stomach" could also mean to put your pen on somebody's empty stomach and write something. If you're going to write on something, you should write on paper. :)