I think the whole body of Tolkien's LOTR-related work makes a good myth.
As for LOTR alone... not so sure. There's no question that it's a good
story, but I don't think it's really a myth. Maybe it's just me, but a
myth seems to be something that's mistier in origin, even when it's laid out as
a tale or set of tales. I think that having read the Silmarillion,
reading LOTR is kind of like skimming the surface of a world full of
tales. And there's also the difference between myth and mythology, which
is what Tolkien was really trying to create (correct me if I'm misremembering!)
That last question is interesting too...
I don't see so much difference between a myth that comes from one man's mind
and one that comes from "cultural fabric". Tolkien was part of the
British culture, after all, wasn't writing in a vacuum. And then there's
the whole question of where this world's myths originate anyway. Maybe
King Arthur began as a tale of old glory days, maybe a single bard scraped old
legends together and created the myth. Who knows!
But for me, the Silmarillion and other old stories are the real
mythology. It's not just a story - it's the logic behind another
world.