Answer to Question 1
He undoubtedly suspects that crying at the drop of a hat is not socially acceptable, especially for boys. Crying about physical pain is more acceptable than crying about emotional matters, and so he is trying to save face in the eyes of his peers.
Answer to Question 2
She provides correct information without making a big deal of his error. In fact, she acknowledges that a lot of children pronounce the word that way, thereby presumably helping Billy save face.
