I used to ride Skyline Drive a couple times a year when I lived in Virginia. I was most familair with the northen-most 33 miles (Front Royal to Elkwallow / Panorama area). Elevation profiles are at:
http://www.bikewashington.org/routes/skyline/index.htm. There's nothing all that steep, but the grades can be fairly long. I'd do it with a 39x26 low, but I wouldn't use even that much (but then again, I love climbing).
Yes, the road is somewhat narrow, and there are no shoulders. Which is no different then any other rural road in Virginia. There were a handful of times I had issues with the traffic -- all but one of those times was when I was on my way
down and I caught up to the cars in front of me. I wouldn't even dream about doing it during leaf-peeping season though. The turns on the descents are banked -- you almost never have to touch the brakes. Just cruise at 40mph the whole way down.
There's only four entry points onto Skyline Drive -- Front Royal, Panorama, I forget

, and Waynesboro. The busiest stretch, traffic wise, is from Front Royal to Panorama. South of Panorama is (or at least was) pretty quiet. I'm not aware of any way to get in without paying the entrance fee; why risk it? It's only $5.
There aren't a whole lot of places on Skyline Drive to get food / water. There's the Dickey Ridge Visitor Center (mile 5), Elkwallow (mile 25), Panorama (mile 33), Skyland (mile 40 or so), Big Meadows (mile 50?), and then there's pretty much nothing. Be prepared!
If you have a chance to do it -- and you like to climb -- you should definitely do it.