Rex Harrisson also played Henry Higgins on broadway. He wanted Julie Andrews for Eliza as well, since she had also played Eliza on broadway. (Julie didn't get the role obviously, but did play Mary Poppins that year, and won the Best Actress Oscar).
Rex Harrison was scared to death of going into a sound studio to record his vocals. He was allowed to sing "live" directly onto the soundtrack, as the film was made. Incidently, Alan Jay Lerner wrote the lyrics specifically to be half sung/half spoken. They were written especially for Harrison, as he was the one and only choice for Higgins when they originally wrote the musical.