Fighting Ring challenges:
Beat the first stage.
Bonus Arena Challenges:
Beat the game to unlock Arena Challenges 41 through 50. Beat those to unlock Arena Challenge 51.
Beat the game under normal difficulty.
Beat Stage 3, then talk to the guy in the shop.
Beat the game to unlock the Carnival, Devil Hand, Karate, and Olivia's Bunny Girl costumes. They can be obtained by starting from Stage 2 then talking to the new NPC in the shop.
Thanks to Revolution Readers Shoni and TheTeh!