Re: Model 1903 Remington
It is a Model 1903A3 made by Remington during WWII. As it is, the rifle has very low collector value; not only is the stock not original (or has been heavily worked over), but the original rear sight has been removed and the receiver drilled and tapped for a commercial receiver peep sight. The barrel date is 8/44; that might or might not be near the date of the rifle.
The USMC emblem is not original and was added after the stock work as a selling point. It does not indicate use of that rifle by the Marines.