Your rifle was not rearsenaled by the military, it was sporterized by someone other than the military. The rifle has been polished and blued, the barrel changed to the later A3 model and the receiver was tapped and drilled for a scope mount. The bolt is an early Remington 1903 bolt (marked R as you noted) that was bent for a scope, again outside the military.
An original Springfield would have a dark parkerized finish, a military rebuild would have a greyish green parkerizing on all parts. In addition, you won't find a military rebuild of a 1903, Springfield or Remington with an 1903A3 barrel. The rear sight for a 1903 would be a ladder mounted at the rear of the barrel where as a 1903A3 would have a peep sight mounted on a dove tail milled into the rear receiver bridge. Thus, if you took a 1903 barrel off and put a 1903A3 barrel on, you wouldn't have a rear sight.
What you have is a typical sporterized 1903 Springfield. It was probably surplused out by the DCM in the late 40's or early 50's. There were many civilians that bought surplus rifles to make sporters and typically they took the shot out barrels off and put new surplus one's on. A3 barrels were in good supply where as 03's were not. The A3 barrels also looked better in that it didn't have the notches in the barrels where the rear sight sleeve mounted and would be exposed in a sporter stock such as yours.
The various markings on your barrel are steel lot codes and inspection marks put on as the barrel went through the various phases in it's making.
Finally, looking at your particular rifle, I would say it was done by a skilled gunsmith that knew what he was doing and did it well.
You could put it back in a military stock but you would need to spend about $300 for a stock and hardware to do it and you would still not be close to original. Add another $150 to have it bead blasted and parkerized and your up to $450, still not original. Buy a 1903 barrel with front and rear sights and your out another $350 or more.
The very best thing you can do is leave it as a very nice sporter that it is and go shoot some moose.