As confirmed by Dwayne McDuffie, the early Thanagarians did not initially have natural wings such as Hawkgirl does now - they relatively recently genetically engineered themselves to acquire them. Three signs point to their non-natural wings here: 1) Hawkman's flying harness, 2) the Thanagarian lawmen in the flashback having metal wings, and 3) the mummies of the Thanagarians at the end having no wings.
The story Hawkman tells is a melding of the DC comics Golden Age and Silver Age origins of Hawkman and Hawkgirl: In the Golden Age, they are the souls of two lovers from Ancient Egypt that were murdered by the priest Hath-Set with an Nth metal blade, and had been repeatedly reincarnated as champions of justice, while in the Silver Age, Hawkman and Hawkgirl were Thanagarian law enforcement officers who were stranded on Earth.
Carter Hall's "real" name of Joseph Gardner is an allusion to his creators: artist Joe Kubert and writer Gardner Fox.
The team of villains the team defeats in the opening scene are the Extremists, "borrowed" versions of Marvel villains who debuted in Justice League Europe #15. They were actually robot copies of super-villains from the other-dimensional planet Angor. In order of appearance onscreen, they are Lord Havok (based on Doctor Doom), Tracer (Sabretooth), Dreamslayer (Dormammu), and Gorgon (Doctor Octopus). Not present is Doctor Diehard, who was based on Magneto.