Is it possible to project an image (like a camera obscura works) 200 feet into the earth in a cavern?

I can see no reason why this should not be possible. You would need a lens with a very long focal length, and that might be a difficulty. Also, the image would be very large on account of the great focal length, and, being large, would also be quite faint, but that might not be a problem in the darkness of a cave.