1)

Tidus: Get [less than] 0:00:00 time in Chocobo Racing (fourth type)
Yuna: Remiem Temple?
Auron: Capture all the kinds of monsters on Mt. Gagazet
Lulu: Dodge 200 Lightning bolts on the Thunder Plains
Rikku: Do the Cactaur sidequest
Wakka: Win it in a Blitzball tournament
Kimarhi: Complete the butterfly sidequest in Macalania

2) The monster arena. Hands down. Capture all the monsters and you'll be able to fight the weaker monster arena bosses. Use the AP tricks to power level

3) Odds are you can. Try it and see.