How important is each stat actually for afflictions?

Engineering increases your Affliction Multiplier, which determines how quickly you inflict afflictions on enemies, but does it also increase the duration of the frozen condition, or the damage over time once they're burning? Or is that determined by the Elemental Alchemy stat?

What does the Elemental Alchemy stat actually do for weapons scaling-wise?

Affliction multiplier from Engineering seems to increase the duration of the effect once inflicted, not the speed at which it's applied.

The speed of application depends on the amount of elemental damage inflicted, which is increased by Elemental Alchemy. Leveling it up increases any elemental damage you inflict, most notably from weapons that have an alternate attack that imbues the weapon with an element for 40 seconds. Alchemy also increases the burn damage from burning effect, but it still deals very little, even if you pump it. As for Freeze and Fulmination, Alchemy does not seem to do anything other than allowing you to inflict the status effects faster. Duration is dependent on Engineering and the Fulmination effect seems to be a static % increase to inflicted physical damage and does not go up even if you pump Alchemy.