Question: When Ray and Charlie are in the bathroom, they sing part of a song. What is the song they sing?
Answer: It's "I Saw Her Standing There," written by John Lennon and Paul McCartney, released November 1963. In addition to the Beatles, the song has been covered by Elton John, The Grateful Dead, Tiffany, and many others I'm sure.
In December 1963, Capitol Records released the song in the United States as the B-side on the label's first single by the Beatles, "I Want to Hold Your Hand."
Question: In the town where Charlie stops to look for a psychiatrist, Raymond wanders off, and is then shown with a juice box and a full bag of Cheese Balls. He seems to have just got them, but how? He doesn't use money.
Answer: There is a deleted scene just before this, where Raymond enters a convenience store and finds Cheese Balls and apple juice, and starts to eat them. The owner asks him for money and he panics, at which point Charlie happens upon them and pays the owner for the food. While he's doing this, Raymond wanders off again, leading to the scene in the intersection. The scene is here on YouTube: https://www.youtube.com/watch?v=IjvvwF5xPyc.
Answer: Charlie probably bought them for him off-screen at some point, since they are his regular snack.
Answer: When they were children, Ray witnessed Charlie being burned by hot bath water. I think Ray says something about how he was left to look after Charlie while Charlie was in the bath, and Charlie got burned by the hot water. That memory must be very strong for Ray, and not comprehending the passing of time, Ray did not want to see Charlie hurt again.
Mark English