So after doing the differing quests in ishgard as a few classes that give you an idea of their military, here is my take on it. Warning, there will be some mild to moderate spoilers and this will be a pretty long post.
The highest power in Ishgard is the church, currently lead by Aymeric and his Temple Knights. From what we've seen so far, the Temple Knights don't appear to owe any allegiance to the four houses, in the case of the 12 knights serving Pope Thordan, they actually appeared to think themselves above the rules and did whatever they want, due to their station no one was willing to say anything. But I don't see that happening with Aymeric though, he appears willing to fight and make sure that his soldiers don't act like that. The Temple Knights appear to be the peacekeeping/police force in Ishgard, the guards we see in Pillars I interpret as each House wanting to make sure that they have their own loyal soldiers guarding their homes rather than the Temple Knights. The four houses also appear to have their own specific designations and duties as well, like how a Fortemps noble runs Camp Dragonhead, Dzemael is well known for building fortifications, and Haillenarte is in charge of Camp Cloudtop.
But there are rivalries in the Houses, during the Machinist story you get a good idea of how the nobles view commoners and the thought of arming them, the main protagonist makes his disdain for them very clear but since he can't directly shut them down, he resorts to sabotage, hostages, and accusations of heresy. Despite knowing full well that he's responsible however, because there is no visible proof or evidence of such behavior, the temple knights aren't able to do anything. So given that, it appears that the soldiers answer to their own House first above all other things as it appears the knights we meet have no issue engaging in any type of behavior (hostages, sabotage, etc.) so long as they are ordered to do so.
Unrelated, the above I feel is a missed opportunity in the storyline and I'm going to mention it here because it still bugs me. The paladin is meant to be a knight that protects the common man rather than answer to a noble house that might ask them to do something deplorable to suit their interests. Having the paladin work on reforming a corrupt knight of some kind could have been a nice theme to explore. But I digress.
We also learn in the Dark Knight story about the Inquisition, the big focus on an Inquisitor hunting a girl. She throws tons of soldiers after you, in the final quest I think you end up killing something like 20-30 of them and even after that she vows to continue hunting the girl at all costs, the Inquisition also appears in the Mch story to address the accusations of heresy. The Inquisition appears to be able to recquisition soldiers whenever they need them as long as they have a legitimate cause on paper, we also learn in the new Hildibrand quest from the Inquisitor NPC that they are starting to lose funding and influence as there have been much less cases for them to deal with.
The Inquisition seem to be the judge and jury in Ishgard society, we see that during the Mch story and the Botanist story as well, they also appear to be very fond of the whole trial by combat thing, but some of them do have some emotion to their behaviors and will consider changing their minds such as what happens at the end of the Btn story where an Inquisitor is so moved by something he appears to change his mind about the whole heresy thing. Malicious accusations of heresy seems to be a popular theme in a lot of the stories, you'd think the Inquisition might start to wonder a little bit?
There are also the common soldiers, part of the Convictory chain quests show some of the common soldiers wanting to kill large dragons because it will earn them fame and recognition. It also provides an interesting incentive, given the nobility's disdain that we see (Mch story) and that capable commoners are elevated and given rewards (convictory storyline) it provides a fairly strong motivation for the commoners that have no prospects in life to join the military. It seems a fair gamble, you can risk freezing to death in the poor sections of Ishgard or join the House militaries and attempt to kill the dragons as a way to earn fame and recognition. Estinien was of common birth (drg story) but because of his skill in combat he was given a great deal of leeway, he appeared to come and go as he pleased and no one really ever said anything to him.
Another thing about the common soldiers you learn is over the course of the storyline for Miner from 50-60, you are given tasks to assess various areas for rare types of ore, but because a lot of the Ishgardian soldiers thought it was "beneath them" they hire a mercenary team to guard the mining expert doing the job (i.e. you the player character). With that it also lead me to the conclusion that if it doesn't involve fighting the Dravanians directly or doing something that doesn't contribute directly to personal glory or standing for their House, the Ishgardians would rather have mercenaries handle it. I was also given the very strong impression from the Mining quests that the Ishgardians regard the mercenaries as a completely expendable resource and that if you and the mercenaries had died in the process they'd just hire another bunch of guys to take care of it.
The Mch story also addresses a lot of the Ishgardian viewpoints on arming the commoners, the nobles have a very strong notion that combat with the Dravanians should only be done by the strong arms of armored knights. The noble sons are also expected to be knights of great personal skill and the only way to earn glory for the Houses is to lead from the front and kill dragons. Proposing something new makes the knights very uncomfortable. Actually, I highly recommend leveling Mch to 60, the story provides a very interesting look in to the views of the nobles about combat as a whole.
That about sums up my understanding of the current Ishgardian military.