3. Be aware that leaving the parkway for camping or services usually means a long steep climb back up to the parkway.
This is commonly repeated, but I'm not so sure. As long as you can get into a campground (Pete's point 1) and carry enough food for a day or two, I think things aren't as desperate as they're often portrayed. For instance, I don't see the climb up to Rockfish Gap from Waynesboro as very bad. Resupply there and you'd want to make the long slog to Roanoke; there's a number of small towns near the Parkway (and not too far downhill), i.e. Floyd and Fancy Gap, from there down to North Carolina. Boone and Blowing Rock are easily accessible from the Parkway. There are got a couple of restaurants and motels around Little Switzerland, then another day to Asheville, and another day to Balsam.
You may be able to plan well and avoid the longer and steeper climbs when leaving the parkway. That said I'd definitely advise planning that carefully and not leaving it to chance. While there are some places where the climb back up isn't bad there are also some real humdingers. Some places leaving the parkway looks like dropping off the edge of the earth. In any case it will be a challenging route for most riders.
Not everyone feels the same as I do, but I really hate going far at all off route for camping, shopping, or other services. I also hate being tied to plan, preferring to be able to play it by ear as I go. As a result I have only ridden the portion of the BRP that is on the TA (Vesuvius to the norther end of the BRP). I have a few times considered riding the BRP and always decided against it after looking closer at the logistics.