It was made by the Aes Sedai during the breaking. You see the scene where they're planning for it in one of Rand's Aiel History Flashbacks in book 4.

It was probably some sort of fold or reality bubble like the Ogier Stedding.

I don't think that the habitat/eye move at all. I think that they're stationary. But instead, the portals to gain entry move to where they're needed. Then again, maybe it did move about and was set down for good after the events in EotW.

I wonder if the grove is still accessible or not now that the eye is empty and Green Man dead and thus its purpose fulfilled.