I was wrong about what can proc Hawkeye, so take my info with a grain of salt. However, this is the Dragon Breath formula:
{[(Caster's HP / 50) + ( Caster's MSP / 4)] × (Dragon Breath Skill Level × Base Level / 150)} × Dragon Training × Elemental modifier(1.25~2) × Ranged Damage(Expert Archer and so on) This is new DB formula.
Taken from here: http://forums.warppo...uildsguide-iro/
DEX affects ATK for bows, but DB isn't a bow, and it doesn't use ATK in its damage formula. DEX won't affect damage. The bit about "Ranged Damage" in the formula relates to ranged damage % modifiers from enchantments (Expert Archer) or equipment, like Evil Marching Hat, Airship Captain's Hat, etc.
Cart Cannon: Assuming the damage formula is accurate, DEX doesn't affect ATK for the weapons that Geneticists can equip, so higher DEX won't increase damage.
Banishing Point: I don't know what the damage formula is, but it's based on ATK. Same here: DEX doesn't affect ATK for the weapons that Royal Guards can equip, so higher DEX won't increase damage.