Sometimes US-276 is closed due to snow or weather, but I don't think it would be closed due to the government shut down. The trail will definitely be open if you can drive on 276.
Unless the road going into pisgah is closed, you can access Moore Cove anytime, that hardly ever happens.
I don't think so. I do not remember seeing any bathrooms or visitor's center, which is what's being affected in most parks. It's just a good hike out and back. It has been raining, so it is probably muddy.
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