Here's my planned route from Chambersburg to Franklin. I have the entire day to drive, so that's why we can take a slower route (my family insists on interstates). . However, I'm taking the fast way back - likely starting north or northeast of Asheville.
Possible diversions:
-Clinch WV 93 near Mt. Storm (not too hard)
-US 19 between I-79 exits 91 and 67 (worth it?)
-US 19 between I-77 exit 20 (WV, near Camp Creek) to Princeton (WV/VA line) - both look like twisty and fun mountain roads
-VA 16 through Mouth of Wilson
-Blowing Rock (cheap, compared to Grandfather Mountain; though maybe hiking up the mountain is possible with time)
-NC 197 near Mt. Mitchell - worthwhile? Would then connect to I-26 north of Asheville, which I've heard is a great drive
I'm sure there's a more interesting route, but this is just looking through Google Maps.
Fair enough 
Asheville is an interesting city. I can definitely recommend the Biltmore Estate tour. They have other activities as well, checking their web site would be a good place to start. I do see that your "listed" age is 16, but I will at least mention their winery tour, with an optional wine tasting at the end. Which I enjoyed.
In the Smoky Mountains, if you are into hiking, you can hike (not too stressful, about 1/2 mile from parking lot at the end of the access road) to the top of Clingman's Dome (on the NC/TN border) which is one of the highest points in the eastern US. There is an observation tower there.
If you are staying in Franklin, you aren't too far from the Bavarian themed town of Helen GA - although that probably falls under "tourist trap".
The natural rock formation Chimney Rock is not too far away.
I'm sure others will have more ideas.
Took a look at Biltmore - yikes, it's an arm and a leg. Will probably save it for another time; sadly, the wine tasting is off the table for this laddy.
I might get a chance at Mt. Mitchell, which is higher - though the view matters more than elevation. I guess we'll have to see.
I feel like I see a lot of Bavarian themed stuff around my part of Pennsylvania...however, I wouldn't mind paying for Chimney Rock (still - $22 per person to walk on a bridge? - no bueno.)
...Man, you're making my "things to see" list longer, huh? That's awesome but sheesh, a week isn't enough.