What does AGI do besides dodge chance? I've seen some runners putting like 9 AGI in some characters. Does it work like DEX, that for every 9 points, it "increases animation for every movement"? Such as one tier of movement speed, animation speed?

Yes, animation speed. It hastens the character's animations ever so slightly, but stacks up on the long run. For swordman by putting 10 AGI it'll unlock an unwanted melee move, so 9 is good. DEX unlocks an extra move of the same type on your combo only in multiples of 10, for AGI every point helps.