So, where's the Sun on the Moon?

[Seraphor]

It's already been explained ingame that Hydalyn is the Goddess of Stasis. In order to keep Zodiark in check, she had to have the astral nature so the moon just happens to be barren, like the First, because her powers lack umbral energies to balance it out.

Quick Google search is not showing anything, but what is the lore cause of Umbral wind?

If the moon is/was housing Zodiark and a gigantic mother crystal equivalent of dark energy, it being the source of umbral weather could be interesting.

We getting our Astrals and Umbrals mixed up again guys?
Astral = Dark = Zodiark/Bahamut/13th
Umbral = Light = Hydaelyn/First