Make sure you have a whole day for exploring Biltmore - the place is huge so theres a lot of ground to cover!
Seafood - mmm... I have heard that theres this restaurant named Trillum that has the best seafood but I have never been there. It is on Hwy 74-a.
There is a guide on outdoors activities here in Asheville - I hope this helpshttp://www.ncguide.com/outdoors/horseback.htm
There are some art galleries in Asheville - you should check out the old Woolworths building in downtown Asheville - they have local artist's works there and its free. Theres another one I have heard about - Blue Spiral, also in downtown Asheville. Ive never been there but I have heard about it.
Just walking through downtown Asheville is great in itself - you can find tons of shops to look through, restaurants, and even just to people watch!