You have a 1903, not an '03A3.
The A3 had a rear sight on the rear of the receiver that closely resembles an M1 Carbine sight.
The 03A3 rifles would be so stamped on the receiver, yours is 1903.
The barrel is a 1944 High Standard, so the rifle was likely rebarreled during WWII Service.
The "P" on the barrel is a Proof mark.
Most other stamps are normal factory inspection stamps.
Workers did inspections during the manufacturing steps and stamped their stamps on the rifle.
"R" stamps indicate parts built by Remington. When rifles are rebuilt or repaired, Ordnance made no attempt to keep original makers parts, whatever was handy was used.
The stock is the so called "scant" stock with a partial pistol grip.
The sight cover is a normal GI cover often put on service rifles to protect the sight.
The receiver and barrel appear to have the standard WWII Parkerized finish, but the floor plate on the trigger guard looks like it's been polished and blued.
The rear sight on the receiver is a commercial Lyman aperture (peep) rear sight added after the rifle was sold as surplus.
These are very nice sights and were a popular addition for hunters and target shooters.