There is one set of mismatch inspection stamps that is correct, that's the one's on yours with a P on the barrel and an I on the receiver. I have seem an I on both, which came before this combination, but never two P's, and never a P on a receiver. All other stamps should match, this type of stamping appears to have started after Savage bought the company in 1920, guns earlier than that will not be stamped there. The combination on yous was used mid 1920's. If I remember right it should have a rubber butt plate, a little later they went to steel.
Get a chamber iron and smooth out the chamber, it might shoot just fine, just don't dry fire it after that. The firing pin should slide out if the lifer is out, the end of it might also be damaged from dry firing and is causing the problem.