Well, while i could go into the history of the 1903 springfield, that is available at any wiki or history site so I'll just ramble a little about what I know when putting a price tag on something like this.
As there is no such thing as a matching gun when it comes to US rifles as they are all refurbished/arsenal repaired after a war, the best you can do is ensure that everything functions and isn't damaged. All rifles are arsenal repaired or refurbed after service so they get stripped down to the screws, the parts dumped into bins, bad parts melted down, good parts fitted to a new gun. So forget "matching numbers," as the Americans weren't big on stamping every little piece anyway.
What you look for are 4 things.
1. is the barrel in good shape?
2. does it headspace properly?
3. does it shoot straight?
4. is the stock/metal in good shape?
From what I see from your pictures, I 'm going to say you have #4 squared away. If the barrel has good rifling, a good crown, no counterbore or bellmouth, and is free of pits, corrosion, and communism, then you can scratch that off your list. You can check bellmouth by putting a bullet into the muzzle, there should be about 3ish millimeters of actual bullet showing between the crown of the barrel and the neck of the case.
Finally, if you haven't already, load up some ammo and see how it prints at the range. A good 1903 springfield is realistically in the 1000$ mark. Mint shape, it can get to the 1200$ mark. The ones you usually see on gun sites for 1500+ are still there because the seller is smoking some really good drugs asking those kind of prices. They stay on those sites forever until somebody dumb enough to pay the price finally comes along and drives the market up. Price it realistically, be honest with whoever is buying it, and remember, unless it was carried by a troop who won the medal of honor and you have the paperwork to back it up, any story that goes with the gun doesn't add crap to the value. You're selling a rifle, not sentimental value.
Hope this helps,
D King