Philip B. Sharpe's book , "The Rifle In America" is probably one of the best references one can find for detailed specifications on American rifles made before 1938 (the year it was published). The NRA Match Rifle Model 1919 (long winded names - even back then) was made from 1919-1933, and was replaced by the NRA Target Rifle Model 19-33 which used the same action but featured a more modern-style target stock and heavier barrel contour.

As for D.O.M., your guess would be as good as mine on that without a serial number (Sav22 on here might be able to help as he's extremely knowledgeable on the early rimfires). Back then S/N's weren't required, but most mainstream manufacturers used them as a tracking tool for future warranty and repair purposes. Is it possible the rifle was refinished at some point in it's life and in the process the S/N was polished off while trying to remove some pitting or scratches in the metal? In any event, I'm betting the 671 numbers on the butt plate and stock are the last three digits of the original S/N. If you had the full S/N the Cody Firearms Museum may have some info on it as they now have all the early Savage record books.