Ease into your tour a bit! Get a motel room close to the airport; most motels have shuttles that will take you and your luggage (including the boxed bike) to the motel. Tip the driver a fiver and you'll get super service. Then you can take your time unpacking and rebuilding the bike in the motel.
(I think there's one decent way out of Dulles, but I've never seen it in a rental car. Dulles is one of the most inaccessible airports I've been into or out of; there's a limited toll access road, what you'd call a motorway, another entrance to the south that looks a lot like another one for the last couple miles in, and if you look really hard at google maps, there might be that other back door.)
U.S. 11 is busier than you might think. There's way more high speed traffic on it than I expected a couple years ago.
Skyline Drive is easily doable in two days (camp at the 30 mile point - Big Meadows?). There's a store and cafe every 15 miles. Climbing out of Front Royal will probably be enough for one day, else you could do the whole thing in a day. Hang your food and you won't have any problem with the bears overnight; they're unlikely to cause any problems during the day.
Rockfish Gap down to Waynesboro isn't bad either way. It's a four lane with a turn lane road that was built before the interstate was completed. Now the interstate carries all the traffic. It's not a problem coasting down or taking the lane on the easy climb back up.