It would help if first you found the date of manufacture
Model 760 D, was just a Peerless Grade - same as model 760 except scroll engraved fancy wood, made 1953 - 1980
Model 760 F, Premier Grade - which is what you have, is the
same as 760 except that it was extensively engraved with game scenes and scroll, finests grade wood, also available with reciever inlaid in gold - add's 50 % to it's value.
Probably worth about $2000 - $3000 in good condition.
I do not know who your Japanese friend was - but he gave you quite a firearm.
That is one nice rifle and a great gift. I was looking at Eugene Myszkowski's book titled " Remington Autoloading & Pump-Action Rifles. I had some questions about rifles I had seen on the internet sites and talked with Mr. Myszkowski about the graded 760 rifles. He informed me that there are a few Japanese engraved rifles showing up. I think you are right when you say the engraving was not Remington. While it has some of the same details there are a few clues that say not Remington. The wood is not right ( you mentioned that); the trigger guard is not engraved; you didn't say if the model and serial number was on the front bottom of the receiver, but I would say it wsn't from the other numbers you gave. I also looked at the gold inlays and found the Remington inlays more detailed and slightly better shaped. There is no question that your rifle is worth a premium but we discourage giving values on the forum. I would get a copy of Myszkowski's book and see what you think.
I was hoping that someone else would answer your question about cleaning the stock. No luck yet. I have been cleaning model 14 stocks for yeras. As long as there is finish on the wood, I like to use Birchwood Casey Stock Sheen & Conditioner. If you try this, first test it on a small area. Use a very light touch untill you see how it reacts with the finish. It will now remove deeplly damaged finish. Many times it is better just to leave things alone. You have to live with your decision. You have also try the cleaner on another gun of less value first to see how you like it. Best of luck.