Actually there's quite a lot of white water kayaking on the Linville River in the gorge. Rather a popular place for some of the serious whitewater types. Check out www.americanwhitewater.org
and search YouTube for a ton of videos on kayaking on the Linville River. Pretty amazing some of the runs they have established there.
But to kayak in? Nawh that's going to be tough. The Falls prevent that and that only leaves some serious hauling to get a boat in there. As far as hunting you'll need to check with the NC Department of Wildlife as far as the hunting regs and season in the gorge. Most people hunt outside of the gorge since hauling game out of those walls ... while not impossible is extreme to say the least. Then there's the fact that it is a wilderness area and is regulated on stay length, group size and the like.
Link for several videos on Kayaking the Linville River