Gae Blog wasn't Haldrith's lance and the design is pretty common among Dragoons as the fins aid the Dragoon in controlling direction mid jump, making their attacks more accurate. Gae Blog actually belonged to a saint, who was a woman, who took up the lance after her husband was killed by a dragon. Cant remember her name though. Regardless, after she slayed the dragon she abandoned the spear and basically took up tending to the weak, sick and poor.

Anyway, Dragoons are an elite unit of knights that technically are part of the temple knights but operate separately mostly, specialising in Dragon slaying. They are rarely involved in other conflicts and such don't tend to get involved in politics. Prior to the end of 3.0, Estinien was their leader. When he got caught up with us Heustienne took command. However she too left command (DRG Job story) so no idea who leads them now.

The Dragoons are a knightly order but that doesn't exclude low born. Estinien himself seems likely to have been low born. I suspect that most lowborn suffer the same challenge getting into the ranks of the Dragoons as any other knightly order though. In both Heustienne and Estinien's cases both were orphans of dragon attacks adopted by former dragoons. Since they and their adopted parents are the dragoons we know best it gives a somewhat limited perspective.

While an elite force I suspect there is a decent number of Dragoons. However they are probably almost all out on missions. You obviously met a few in the DRG story quest both prior to and in HW.

As for the rest of Ishgard's military, you have the Temple Knights which serve the Church directly and are under the command of Aymeric. In addition to that, each High house has its own standing forces drawn from their ranks and any lower houses that serve under them. The Holy See places responsibilities for particular missions and such on each house and those houses can likewise request aid from the other High Houses. Since the ability to provide defence of Ishgard greatly effects political standing the high houses take their ability to take those responsibilities very seriously. Failures, such as the fall of the Stone and Steel Vigils, can have very bad political ramifications.

The Lowborn hold no standing armies traditionally. Slaying a dragon can earn a Lowborn standing and Lowborn can be recruited by the houses as leves. That has changed somewhat recently with the rise of the Machinists and with the establishment of Hilda's watch.