Nice liking rifle.  Getting back to part of your original question...yes, that wood would probably respond nicely to some "gentle" cleaning.  There are as many ways to do that as there are collectors.  There is no absolute "right" way, but very many "wrong" ways.  If it were mine, I would take a pad of 0000 grade steel wool and dip it into a bowl of turpentine.  Place the stock muzzle-end up on an old towel.  Start rubbing the stock near the muzzle gently with the wet wool making sure that you keep the wool and wood wet with turpentine.  You should notice gunk dripping down the stock onto the towel.  Keep working your way down the stock being very careful not to rub too hard, especially around the markings.  You want to remove dirt and sweat and oil--not wood.  You might need to do this two or three times to get it really clean.  Then, wipe it down nicely with a rag soaked in turpentine.  Let it dry for at least 24 hrs. and then apply some BLO (google how to apply BLO, it is super easy).

daveboy