IE: Battle Red, Red battle music. Battle Cynthia, Cynthia battle music.
Yes it does
Battling in the Champion Cup will have all of the champions, plus their music. Win-win situation! If you also battle Cynthia in Undella Town, her music will be played aswell
All of these people plus their music are in the Champion's Cup. Even though Alder and Blue aren't the champions of their regions anymore....