Is there an S stamped on bottom of magazine well, under where the floor plate sits? (indicates Springfield Armory mfg)

Yes, there should be a pin in the hole. If it is missing, make one from a wire brad or perhaps a clothes hanger. As mentioned above, with a USGI trigger installed, the rifle can be fired by pushing the trigger forward. This was a design/manufacturing defect in early M1903 trigger guards, ca. 1905. This was corrected in newly made TGs by shortening the slot.