The music played during the scene where Gene Hackman gives Sigourney Weaver the gift of a lighter is the same classical piece that was used as incidental music in Weaver's debut film, Alien (1979) (it's playing while Dallas is sitting alone relaxing in the shuttle). This has to be an in-joke intended by the filmmakers.
When Jack joins Max in her hotel room she intentionally leaves the door slightly open (in order for Page to be able to enter later), but when Jack wants to leave a little while later, the door is suddenly closed.