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This started as just a hunch, but now I'm convinced it has to be Star Ocean: The Second Story.
1) Your serpent-wearing, barrel-obsessed character is Ashton Anchors
.
2) The girly-named character is named Precis
.
3) There's a special skill called "Come on Bunny" to, well, summon a giant bunny as transport. Here's a video
; skip to 1:20 for the bunny in action (everything before that is the menu showing off characters' voice clips).

A videogame I played as a little kid so it's probably a bit older. I just remember random bits and pieces from it. There was an arena you could go to and fight different villains and characters to win points in and there were a variety of characters you could play but the arena wasn't the main point of the game. There was a character who had dragons or snakes attached to his shoulders who was obsessed with barrels, and a girl named Precious or Patience (something girly like that) but I can't remember much of the characters other than that. There was also this bunnylike animal that could carry you from place to place in its pouch. I'm also pretty sure it was a Japanese-made game. It had pokemon like graphics.