Are songs different between subbed and dubbed?
Sachi in particular hummed the story in the Japanese version as well. (Though, in the book, she actually sang "Rudolf the Red-nosed Reindeer". For whatever reason, they altered that scene in the Anime)

In general, what you're saying will vary from anime to anime, and it depends on who is dubbing them. How much localization the studio wants, I'm sure is a big thing when it comes to this. "How much of the origin language's references do we want to keep?"
Overall, it'd difficult to directly translate a song from Japanese to English, while keeping the flow and meaning intact.