One of the songs that Nelly sings is "The Flower of Gower Gulch," first sung by Porky Pig in the cartoon Drip-Along Daffy (1951), which was also directed by Chuck Jones.
The magazine "Wife" is a parody of "Life" and features giraffe neck sweaters. A turtle peruses this photo spread and sadly remarks, "That's show business!" which is a play on the turtleneck garment being around first.
As Nelly leaves her friends, the music playing is "Aloha 'Oe." When she sails into NY harbor, it's "Columbia the Gem of the Ocean."
Nelly's "scandal" is a direct reference to Ingrid Bergman, who was shunned by American audiences in 1950 for her (at the time) scandalous affair with married director Roberto Rossellini. Bergman spent most of the 1950s away from the US, marrying Rossellini in 1950 before divorcing him in 1957. After winning an Oscar in 1957 for Anastasia (1956) the scandal was largely forgiven by the public.