Before I answer your questions, I'm going to have to hit the brakes a little and point out that this part of your post in the IC thread:
the Battle mage was caught off guard as the lower ranked knight all of a sudden sprang on the superior
and this:
blast of air aimed at the attacking Leo.
... didn't happen. While Leo is certainly challenging Tyrhallan to a sparring match, he hasn't actually attacked yet, so would you mind editing your post so it doesn't seem like Arn is blasting air at Leo for no reason? I'm sure it's a simple misunderstanding, but please try to avoid jumping the gun again in the future as it technically counts as breaking rule 1. Do not
control, maim or kill anyone else's character(s) without their consent. I don't mean to put you on blast, I just have to point it out as the gm. I hope it's not too much of a hassle.
Now to answer your questions.
am I to assume that we are following military ranks?
Sort of. If you're familiar with Warhammer 40k, think of it as how a space marine chapter is organized. It'd go from soldier > sergeant > captain > commander > general and of course the king would have the highest level of authority, but it wouldn't get much more complex than that because there just aren't enough soldiers to justify it. Sure it's been a hundred years since the cataclysm, but the world's population is still a lot smaller than it used to be, so there's probably no more than a few thousand soldiers in any nation's army. More than enough to man the guns of an airship though.
Would knights and mages be in the same service Army perhaps?
Knighthood is a rank in of itself and stands at the highest among the army, as described in the lore. The only ones they answer to is the king of Belisio, the general of the army and their Knight Captain, Tyrhallan. Battle mages on the other hand are more of a role than a rank. They're mana specialists trained to engage in direct combat or advise whoever's in charge wherever they get assigned to, either in teams or individually, so they're more of an attachment rather than a part of the army. Same thing for casters.
would battle mages hold a rank based on skill or are they only based on birth?
They don't hold ranks among the army, but they would among themselves based on seniority and leadership. All mages receive their assignments from the Archmage. Hope that answers all of your questions. If there's anything else you want to ask, please don't hesitate. It helps me just as much with the worldbuilding.