Springfield Armory (the real one, not the current commercial one) couldn't keep up with M1903 production during WWII and Remington and Smith-Corona both made them and a combination of having to re-tool as tooling wore out and ongoing modifications to simplify production and reduce costs lead to the M1903A3. There are plenty of websites where a serial number search will date the original rifle (to either a specific year or a year range) but as a general thing here's what you've got on your hands.
The rifle is more than 65 years old since Remington started production in 1941 at serial number 3,000,000(?) and I think they stopped production no later than the end of the war. Your rifle was sporterized 50-60 years ago and the term "professionally sporterized", which covered a lot of ground at any time, includes everything from masterpieces to dangerous junk.
Likely, if you know and trust the seller it is at least safe to shoot but just as likely you've got no real idea of the condition of the rifle. As a minimum, I'd have it checked by a gunsmith or knowledgeable collector who is familiar with them. Have it checked with go, no-go and field gauges and, if you can find them, throat and muzzle erosion gauges and be darn sure to have the trigger and bolt assembly checked since many have been monkeyed with over the years. It is worth having the uniformity of the bolt lock-up checked since a lot have had either enough use or not-so-good modifications as to not be uniformly bearing the load on the lugs.
These things were made with a lot of different barrels. Rifling ranged from 2 lands and groves on up and all types were capable of shooting great groups but some were pretty sloppy shooters. All saw corrosive primers and the general condition is often related to how religiously it was cleaned after firing.
You'll to shoot it to see what it likes best but they tend to like 150 to 175 grain spitzers at moderate velocities. If you reload you have a better chance of getting very god groups but the only way to tell is at the range. If the scope mounting does not block the stripper clip slot you can get five round stripper clips from any competitive marksmanship supplier (Champion's Choice, Shooter's Choice etc). If the scope is over the center line of the bore you'll be single loading.
Good luck with your new best friend.
Lance
