Breath of Fog

More so on Y!Tiki than A!Tiki, I'd say. With DC, you'd want to bait enemies. Since dragons and mages both target resistance, you'd want your bait unit to have higher resistance, and A!Tiki's resistance is just not good enough for that (and the +Eff forge doesn't help in that regard either). It could work and would be good counters for those, but you'd also be sacrificing a Breath skill by using DC. That accelerated cooldown is just too good to pass up, especially since Lightning Breath comes with DC built in already. Might as well just use Lightning Breath, tbh.

Yes, it has all the effects of Falchion: Effective against dragons and restores 10 hp every 2 turns. It also has the adaptive damage of breath weapons targeting the lower stat of enemies with an attack range of 2 on their weapon.

Breath of Fog does not have built in Distant Counter, so you would have to give your dragon the Distant Counter A skill.

So, in short, yes it does extra damage against dragons and will target the lower def/res stat of mages/archers/daggers.

I like the potential of this build more, but it is much more situational. The heal effect is good, but this doesn't replace the overall usefulness of Breath skills, unfortunately.