Old Chestnut

Max has a rough day and gets detention for claiming a dog ate his homework. His friend explains the term 'chestnut' as an old, overused joke, tracing its origin to a play by William Diamond. Max then reveals that a dog indeed ate his homework, prompting them to talk to his teacher.

What story did Captain Xavier tell in the play mentioned?

How did the term 'chestnut' evolve to mean something that has been heard many times?