Hi!
I do quite a bit of work in reproducing historic bags made from hair-on animal hides, and I rarely see edges left exposed on originals. Saddle-stitched inside out, as Somawas has suggested, seems to be the most common.
One method that has not been mentioned yet is edge-binding with leather. Thin sheepskin or kid can be sewn over the seam edge with the same stitching that draws the body together. I use a normal saddle-stitch but I presume more decorative techniques can be used as well. The hair will still catch in the stitching if it is very long, but in most cases you shouldn't need to trim.
The binding leather can easily be dyed beforehand to match the hide, or to contrast, or can be left natural.
The other (very) sneaky advantage of an outwards-edge-bound seam is that you can affix a lining fabric to the inside of the hide parts and then pinch the lining into the seam at the same time sewing both at one go.
Undyed sheep binding around the outside edge of a knapsack flap:
I hope I've explained it right. I think like most on here I'm better at the doing than the talking part!
Good luck with your projects!
(P.S. Sometimes older books have techniques the new ones leave out. Somewhere I have a How-to from the 1940s which was mostly Celtic-influenced work and cut work, inlay work, and as you mentioned setting glass and semi-precious stones into leather -- 'Ruskin Stones' as it called them. never done it myself, but I'll see if I can rustle up the book...)
