For cleaning up the metal parts, START with extra-fine steel wool. It is hard enough that if you add elbow-grease, it will cut surface rust down nicely. Any heavier rusting, use FINE steel wool. This is a 1903: super rare. Don't go any heavier than Fine. Then you wipe it down with a rag, then again with an oily rag. Remaining blue (and it looks like a lot there) will GLOW.
In your PM, you mention the chamber has very light rusting. Try a twist of the Extra-fine steel-wool on a cleaning rod. Insert and twist, running it in and out. Put on a fresh twist and see if any rust comes. If not, then you oil the chamber lightly, then wipe it out. For the bore, just a bronze brush should do the trick, with some powder solvent. Oil and wipe, leaving just a thin film of oil. Clean out with a dry patch, bore and chamber BOTH, before shooting.
Missing screws: should not be a problem. Many of the parts on the 1903 were contract-made in the USA, quite a number by Frank Mossberg. All Ross Rifles use standard American threads (60-degree V threads), although the threads for small parts will be super-fine. No real matter: any machine-shop in the country can make one, once you know the size and pitch (which I do not for the Bolt-cover Screw: anyone here know it?).
The spot where your adjustable Rear Sight attaches to the Rear Sight Arm is quite vulnerable. I have a 1905 here with the same sight, made not long after your rifle was made; it has been broken and soldered at some time. I would suggest having yours silver-soldered, as this will form a very strong bond which is not likely to break again. At least you have the PART!!!
The Stocks on these were Walnut and should not be TERRIBLY heavy, although Walnut does vary in density quite a bit. A big problem with military rifles of this age is the fact that the Army liked things to be OILED, which, all too often, meant extra oil running from the metal, into the woodwork and being absorbed by the wood. To CLEAN the stock, try warm water and a good detergent. Paint splashes and things like that, often discovered on old rifles, may be removed with gentle pressure from the blade of a flat screwdriver, then the area smoothed with extra-fine steel-wool. This will leave some marring to the finish which is there. Once you have done this, determine if the wood is oil-soaked or not. I have had decent luck removing motor oil from old wood with Brake-kleen. When no more motor oil comes out, you can start in to get some of the RIGHT oil into the wood. REAL Boiled Linseed Oil is what the wood wants. You soak a rag and wipe it on, then wipe it off. You don't want a shiny finish, but you do want the wood to have a drink. The BLO will set up inside the wood (in time) and protect it from further invasions by engine oil. Once the wood has had its drink, it will just sit there and quietly glow for you. These were truly a beautiful rifle when they were made.
Treat it like a 105-year-old baby, which is what it is..... and almost as rare.
The fore-end has been chopped, as you can see. The GOOD part is that the chop is under the band. Very little trouble to make a new fore-end and attach it. The HARD part will be finding a Nose-cap. Usually they used straight-grained Walnut for their rifles but there might be a difficulty in a perfect colour match: Ross bought their early wood from France, Rumania and Italy. Oh well, that's why God created stains! Try at least to get the grain right, then stain to match.
You have a FINE and a VERY Rare Toy. Some guys are just, plain, born lucky!
Hope this helps.
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