Not that simple a question. To briefly answer just one aspect, many jazz performers also were trained in classical music, so the idea of having structure in music (from the classical genre) mixed with the idea of independence and solo performances (from jazz and blues).
That mixture was one factor in creating the musical form heard on many pieces in the 1930s and 1940s, where songs consisted of strictly arranged stanzas punctuated by improvised solos.
To name a few performers, Fats Waller studied Bach's music and released a number of performances of his organ works. In addition to being the King of Swing, Benny Goodman was one of the foremost interpreters of Mozart's clarinet pieces during the 1930s and 1940s. Duke Ellington used classical forms in many of his longer pieces. Woody Herman and Igor Stravinsky collaborated on experimental jazz-classical fusion pieces, and at the time of his death, Glenn Miller was studying classical arranging and conducting in England.