I doubt you will have any trouble accessing the trails. However they will close the Blue Ridge Parkway if they see any chance of icing. I have use this number for BRP weather conditions 828-298-0398. BTW this is the termination of section 12 at Beacon Heights if my memory serves me correct.
This section passes through the Wilson Creek (proposed wilderness area) which is really nice. I did my first hike of this area last year. Not sure how I avoided it all these years, but this will definitely be one of my winter locations. The elevation of the area is lower, so no real worry of bad weather. Check the forecast before going, but I would guess things would be open over 90% of the time. Lower trailheads around Wilson Creek would be easy to access.
When I did the area I just pieced together several trails and made a 2 day loop. Lot of options if you would rather do a loop as opposed to section 12 which would require a car drop or some help shuttling.
I have a map of section 12 with some camp sites highlighted. Can't comment on them, but Wilson creek has plenty of camp sites. From a quick look at the map I would say you would have a ton of options to chose from. If you would like the map send me a PM with an email address. It is in pdf version. Only limitation in camping might be close to the BRP.
Let me know if you have more specific questions