Ash of Agni and it's Alchemy Path

While the dank vape is out, does it only inflict fire, or does it also damage monsters inside the mist?

It only sets enemies on fire, but you can re-inflict a status, and you take damage every time you catch fire. Fire ticks are every 2 seconds, but haze ticks are every 1 seconds. For this reason, enemies take damage faster inside of a haze then from just being on fire.
Overlapping hazes onto an enemy will damage them even faster

Do the haze ticks do more damage as the bomb levels up, or is it the same fire damage throughout all tiers?

The fire damage is the same, lvl only affects the dmg of the bomb itself, and the ctr (at lvl 5, 10).

Fire and shock does damage in tiers, and fire from a tier 1 source will be very disappointing against a tier 3 enemy.
However, the fire damage from 4* Fiery Atomizer will be identical to 5* Ash of Agni regardless of depth or heat level, and a 2*/3* Fiery Hazes will be identical in damage with 5* Ash of Agni in T1 and T2 depths.
explodes and does elemental damage around the bomb, then incinerates a big area doing fire damage and inflicting fire