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1) You are correct - the barrel and receiver are original to each other.
2) The stock MAY be original if there are not any 03A3 handguard retainer ring "slots.
3) The bolt is probably a replacement. All Remington bolts had an "R" under the bolt handle. First see if the bolt handle is "straight" or "swept". Also look carefully on the top of the bolt handle and on ALL sides of the bolt safety lug (near the middle of the bolt body)
4) The buttplate is a replacement. Remington M1903 buttplates were "smooth".
5) Does the rifle have stamped or milled parts - look on the upper and lower bands, the trigger guard and swivels. Answer that and I can tell you where to look next.
6) Look on the rear sight - right side. There should be R markings on the windage knob, rear sight leaf and rear sight moveable base - also possible on the sight binding screw.
7) Assuming the upper band is milled - there should be an R on the right side.