This rifle was made in June 1943. The original barrel should be dated in the same month, or perhaps up to 2 months prior or 1 month after the receiver was produced. The barrel could be either 2 or 4 groove as both types were produced at the same time during this timeframe. The stock has the metal reinforcing pins, rather than the screws which were used in later production. The trigger guard is also the early "short bow" style. The transition dates for the trigger guard and stock styles aren't precise, so either type would be correct for this vintage of rifle.
Original Remington 03-A3s only had the receiver and barrel parkerized. All other metal parts were originally blued, although the upper bands showed up parkerized towards the end of production in early 1944. Virtually all metal parts, save for springs, screws and pins, should be stamped with an "R" on Remington made 03-A3s.
This rifle appears to have been refinished at some point as all visible metal parts are parkerized. The metal finish is also quite pristine vs the condition of the wood. Many 03-A3s were reparkerized during military overhauls. Many would also show misc replacement parts produced by Smith-Corona or other parts contractors that were swapped in during the repair/overhaul process. If the rifle went thru military arsenal overhaul the stock would show the marks of the arsenal that did the job. I have owned all of the Remington made 03-A3 parts in NOS condition and all were blued. In doing restorations I always try to locate parts with an original blued finish, although reparkerized ones are every bit as serviceable.
Value depends on condition and originality. This rifle appears to be in nice condition, but the metal parts do not have their original finish.