Sounsyy pretty much covered everything known about the Sultansworn and the PLD Job, so all I can really add is pretty much speculation about how their abilities are powered (thus answering your question about whether they use aether or not). Firstly, they can definitely make use of basic Conjury spells like Cure and Protect to aid them in battle, meaning some form of aetherial manipulation is required, but more than that, some actual PLD abilities clearly are manipulating the aether of the PLD themself - Spirits Within is an example of this, it is directly connected to how much HP the character has, a gameplay analogy for lifeforce. Although Spirits Within does not consume said life, it still requires it to function.

Then there is the level 58 ability Clemency, a powerful healing spell. This is clearly 'holy' in that the PLD kneels down as if praying when the spell is cast, and what appear to be angel feathers materialize around the PLD when it's cast. So there is definitely 'light' aspected aetherial manipulation going on, in keeping with the PLD Job's traditional roots throughout the FF series, as 'holy' knights who can also use White Magic (and the natural antithesis of the Dark Knight).

Anyway, good work Sounsyy for that in depth description on the PLD Job.