In terms of question 1, did the owner or architect of Sand Hills copyright their work ?
In the US, you don't have to copyright your work. You own the copyright when you create it. That said, copyright might not even be the right area of intellectual property law. Perhaps trademarks would apply instead, I don't know.
In any event, this is why I'm hoping someone in IP can better explain this to us, because it's still clear as mud.
That's true about copyright everywhere I think. If you create something, it is your copyright. It doesn't need to be asserted.
It's true in most places. There is, I believe, a registration system in the US (caveat - I'm a UK IP lawyer, not a US one) but registration is not a requirement for subsistence.
I don't really have time at the moment to go into detail on this but there's some fundamental misunderstanding of copyright being shown by several people expressing strong opinions on this thread. I don't see a problem with the concept of copyright subsisting in an original work of golf architecture in the same way that it is already clear in the UK that it can subsist in an architectural work (s4(1)(b) Copyright, Designs and Patents Act 1988 provides that "a work of architecture being a building or a model for a building" is an artistic work). There will not be copyright in elements of a design which are not the architect's original work.
The suggestion that copyright is designed to protect industries is historically and factually nonsense. Intellectual property law is never perfect because it is intended to attempt a fair balance between the protection of creative and inventive effort and open competition and the balance changes as rapidly as the world changes but, actually, legislatures haven't done a bad job of maintaining a pretty good balance.
I have advised on a situation where a course tendered work, chose an architect and then asked that architect to build to another bidder's proposed plan. Do those here who dislike the idea of copyright in golf course design think that's OK?