Fire III is a little awkwardly timed in its introduction.
The thing that affects the fire stance damage buff and the ice stance MP refresh isn't which spell you cast but how many stacks (floating balls) you have.
Umbral Ice I: Fire spells are 0.9x potency. Refresh 3200 MP per tick.
Umbral Ice II: Fire spells are 0.8x potency. Refresh 4700 MP per tick.
Umbral Ice II: Fire spells are 0.7x potency. Refresh 6200 MP per tick. Fire spells cast twice as fast. After Lv72, Fire spells cost 0 MP.
Astral Fire I: Fire spells are 1.4x potency and double cost. Ice spells 0.9x potency.
Astral Fire II: Fire spells are 1.6x potency and double cost. Ice spells 0.8x potency.
Astral Fire III: Fire spells are 1.8x potency and double cost. Ice spells 0.7x potency. Ice spells cast twice as fast. After Lv72, Ice spells cost 0 MP.
Fire I and Blizzard I build stacks one at a time. Fire III and Blizzard III immediately put you at max stacks. However, when you first learn Fire III, you only have a 2 stack max. So even though you learn Fire III at Lv35, it's not worth using until Lv40 because it takes so long to cast that you might as well just Transpose and cast Fire I.
Once you hit Lv40, you can stack 3 balls and your Fire and Ice spells get a 50% haste buff under three stacks of the opposite element. Then Fire III and Blizzard III are the best way to switch elements in single target.
When syncked down, you won't have access to any actions that are too high for the sync level. As far as how damage scales, you can always compare potencies to each other without worrying about damage scaling. A spell that does 200 potency is twice as powerful as a spell that does 100 potency whether you're Lv1 or Lv80. Just remember to take into account the potency adjustments from Astral Fire / Umbral Ice, which take effect without changing the potencies on the tooltips.
All of this makes BLM one of the most complicated jobs in terms of how its rotation changes at various levels. Rather than having one or two big evolution moments like other jobs, it has like ten different milestones where the rotation needs to be adjusted.