I would refinish the stock using Boiled Linseed Oil, easy to apply, probably what was on it to begin with, and looks great.
Go with the barrel code and ignore the serial number "range" thing because they are just a guide and it was common practice at Remington to make rifles outside the serial number range during any given month. I think you will find that the magazine holds five "5" rounds and not six.
As far as the different serial on the stock & butt plate, again this wasn't an unusual practice at Remington. I mainly work on Model 14's and it took them more than 25 years before they started putting matching numbers on the rifle, stock, & butt plate. It could also be the rifle was restocked at one time but rather doubt it, they're very sturdy, and rarely if ever broke unless you drove a truck over it.
I broke form replying to this only because I'm an "old" small bore marksman that appreciates the rifles made in that era but there could be someone else on the Forum that could provide you with better information.
Jim Peterson
Charlotte, NC
New member with 513T
Re: New member with 513T
Jim Peterson