Vigor effects the characters attack, so the higher it is the more damage they'll do with weapons. Magic effects magic, obviously. The higher the number, the more hp healing magic will recover and more damage attack magic will do. Speed effects how fast the ATB bar fills up. Stamina, as far as I'm concerned does absolutely nothing. It's supposedly increases the max hp more when a character levels up but, I never notice the difference.

Yes, the Atma weapon is based off the characters hp. The higher the hp the higher the damage.

The Assassin weapon does randomly dispatch a monster, however, if the monster is undead it won't be effected. That's why sometimes the monster disappear and then reappear.