The scene featuring Robert Morse skipping & dancing down the street on his way to work (immediately after the "Old Ivy" fight song duet with Rudy Vallee) was filmed on location in New York City using hidden cameras and a small earpiece to cue Morse on his timing. The various amused & astonished passersby were not extras, but rather were New Yorkers reacting genuinely to someone dancing to his own tune.
All of Rosemary's songs (including "Happy To Keep His Dinner Warm" and "Paris Original") were cut from the movie version. To make up for this "I Believe In You" was given to her for the movie. In the stage play, she does not sing this to him, and the first time it is heard is during the scene where Finch sings it to himself in the executive washroom, but she does a brief reprise of the song after this scene. In the film, she sings the full version in an earlier scene, making Finch's washroom version the reprise.
The original Broadway production of "How to Succeed in Business Without Really Trying" opened at the Forty-sixth Street Theater in New York on October 14, 1961, ran for 1417 performances and won the 1962 Tony Awards for the Best Musical and Book and was nominated for Best Score. Robert Morse (Winner of the 1962 Tony Award for Best Actor in a Musical), Rudy Vallee, Ruth Kobart and Sammy Smith recreated their stage roles for the movie version.
In 1964, Tony Curtis expressed interest in playing the role of the up-and-coming young business exec eventually portrayed by Robert Morse, even though he was then nearly 40 and far too old for the role.
A self-help book titled "How to Succeed in Business Without Really Trying" did exist. The full title: "How to Succeed in Business Without Really Trying: the Dastard's Guide to Fame and Fortune" by Shepherd Mead (New York: Simon and Schuster, 1952). At the time-of-setting of the play and film, Simon and Schuster had almost certainly issued it in mass-market paperback, so that a real-life J. Pierpont Finch could indeed have picked it up at a kiosk and run with it.