So the images and the details of what you have, were edited out as something designed post 1865 exceeds the parameters of the forum, BUT..., since rust and wood finish are a problem for both antique muzzleloaders as well as reproduction rifles and such, I left the basic question up.
Taking care of the rust beyond just dowsing the metal receiver, barrel, and hardware with a rust preventative which will stop the rusting, often means that bluing or browning will also be removed. This often drops the value of the piece.
Refinishing the stock finish is the same story. IF you strip it down, re-stain and refinish, you are likely to remove patina that adds value.
I'd suggest wiping down the entire piece with Birchwood Casey Barricade, to stabilize it, and then take the gun to a firearm restoration person. Actually I'd likely take it to two that are independent of each other, and compare what they say. Just as I'd get a "second opinion" from doctors on a serious medical issue.
LD