Do they get a notification informing them that the question is closed?
At present, they do not (see In which way the OP is notified about close votes?). There is an old and popular feature request asking for this to be changed: Send authors an inbox message if their question gets closed.
Usually at least one comment is left before a question is closed, which serves as a kind of notification. When I review a question in the Close Queue and there are no comments yet, I usually post one.
Still, this isn't a satisfactory state of things. I don't know what's stopping SE from sending a notification of similar style as for expired bounties. For what it's worth (which is nothing) I put a bounty on that feature request.
What does a user see when their question is closed?
Almost the same thing you see. For duplicates, the language is a bit different: it mentions editing the post rather than asking a new question. And the banner on top of the question is visible to them sooner, after the first duplicate vote.