I would presume either Conjurers or Alchemists can treat people for Aether sickness, since both hve strong medical backgrounds and it is clear treatment is possible.

Most Final Fantasy games have a force that fuels magic, be it Mist, the Lifestream or so on. In this case, it's Aether. Eorzea's potent aether can be seen as a parallel to the Mist Continent in FF9.

The Lore forums have discussed some interesting theories about Aether, including the notion that it's what we Earthlings know of as 'dark energy', that it gave birth to the Twelve in the same way it does the Primals and so on.