Arturo Sandoval and Dizzy Gillespie did in fact perform together as depicted in the film and it was Gillespie who helped Sandoval and his family defect and the adjustment to being in the United States soon after.
Most of the songs heard in the film are in fact Arturo Sandoval originals. They've been featured on Sandoval's albums throughout the years.
Singer Gloria Estefan, who plays Mia Maestro's friend in the film, has in real life had Arturo Sandoval play on some of her albums.