I'm pretty sure there is no legal requirement to tip. However, if wait staff end up below minimum wage after tips their employer is supposed to pay the difference, though this usually doesn't happen and most people aren't aware of it. Wage theft is the most common form of theft in the US, and this plays a part in it.
I'm pretty sure this wouldn't count as counterfeit. It's not meant to pass as real money. It's gag money. You're not required to tip though, so there's nothing illegal about not leaving money or leaving something else instead. Obviously knowingly using counterfeit money is illegal regardless, but again this is counterfeit, just a gag.