I have been up to the top many times and never paid a fee. TN 143 is a state road that passes through the park and takes you into NC. Therefore, you can drive it anytime, day or night. At the top of the mountain are several parking areas. You can hike up through the Rhododendren Gardens to the Appalachian Trail. There are no toll gates.
I don't believe there are certain times you can or cannot enter the park. There are no gates you have to pass through to enter unless you go up to the very top of the mountain to hike the Cloudland trail which is actually in the Cherokee National Forest. There's a small fee of, I think, $3.00 to enter this area.
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