It is indeed a carpet anemone, Stichodactyla tapetum. They're commonly called mini carpets because they only grow to be about the size of a half dollar. Feed them and they'll split like mad.
Just wondering, how much is your LFS charging for one?
I have two of them sitting in the middle of a patch of palys and they don't seem to hurt them at all.
No they won't host clownfish.
As far as the lighting, I would say yes.