S/N 3918617 is an M1903A3 rifle. The receiver should read: U.S. Remington Model 03-A3 3918617 The M1903A3 rifles were a simplified version of the earlier M1903 rifle primarily made by Springfield Armory and Rock Island Arsenal. Many of the metal parts were changed from milled to stamped to speed up production, and the complicated rear leaf sight on the front of the receiver was changed to a simpler, easier to use peep sight mounted on the rear of the receiver. Remington produced these rifles from late 1942 up until February or so of 1944. Your particular rifle would have been made in September of 1943. Value is impossible to determine accurately without good clear pictures of the rifle at the very least. There are a number of factors which would drive this - whether the rifle has been modified from its military configuration (i.e. 'sporterized'), whether the rifle has its original Remington parts, whether the rifle has been refinished, etc. An unmolested, original finish Remington M1903A3 in good condition will be worth around $750-900. A badly-sporterized M1903A3 in fair to poor condition might be worth $100, assuming you can even find anyone interested in it.
RW is Remington Works, not part of the serial no. 1912.
Manufactured 1912, history none is available.
Remington's website has a history section that you can look at.
1914
What caliber is it? The length of the rifle cartridge determines if it is a long or short action.
Go to Remington's web site. They have a section that describes their products, present and past.
most likely it is with 22-250 rounds.
1914
General information on the 721 can be found on the Remington website here: http://www.remington.com/library/history/firearm_models/centerfire/model_721.asp
around 1912
Go to remington.com.
Contacting Remington would be a long shot, and it probably wouldn't tell you much about the rifle's history, since it was a military weapon. What you have is probably a P14, manufactured under contract for the British during WWI.