Typo: The "Atlantic City Daily Press" subheadline reads "Five Women Have Come Foreward" (misspelling of "forward")
While not exactly an anachronism (Vivaldi's Op.10 Concertos were published in 1729 and were known in 1921), his works had fallen in oblivion, and were unknown to general public, except for some experts and musicologists. After the revival of baroque music, the first known recordings of a work by Vivaldi were made in the early 40s -the now infamous "Four Seasons"- and therefore is extremely unlikely that Gillian could dance for the Commodore to the sound of one of Vivaldi's flute concertos.
The depiction of Benny (Bugsy) Siegel is not entirely correct. He was called "Bugsy" because he had explosive temper which could erupt at the slightest reason or for no reason at all, not because he, as in this and every episode he appeared, would do nutty, crazy stuff.