Shor = Lorkhan.
Lorkhan = the moons.
Gods = their realm.

Therefore Lorkhan's realms are the moons, which means that they're Shor's realm, meaning Shor's realms are the moons.

But isn't Nirn also part of Lorkhan? (Or rather, isn't part of it also Lorkhan, as well as the aedra, thanks to it being powered in part by his heart.) By that standard, Sovngarde is just the bar down the road.

I think that Sovngarde is Aetherius for the Nords who die valiant deaths and who lead noble lives. For Altmer, Aetherius would be something else and for Bretons it would be another thing. As for the Dreamsleave, yes, souls might be recycled, but you might still remain in Sovngarde. Maybe you can seperate your life force from your soul? I am not sure, and I don't think it has been explained or ever will be. But what I know for sure is that there are some pretty damn old people still hanging around in Sovngarde who haven't been recycled yet (Ysgramor, Olaf One-Eye, etc.). The entire process is (intentionally?) unclear.