Even before you posted the GI link, I believed your rifle was likely built by Schüler from the features, so I would agree.
Stmg is in relation to proof load, meaning Stahlmantelgeschoss or that it was proofed for steel jacketed bullets, and should give the bullet weight as well. It has subsequently gone through a British proof house and is reblued so it maybe tough to see additional markings. Below the wood line you should have the additional German proofs which will give you more information. The bolt jeweling and checkering isn’t original, but it is still quite a nice rifle, with a lot going for it. The pad looks to be original to it, and original rubber pads aren’t super common this early on, but this would be the kind of rifle they show up on. Schülers in 11.2 and 12.7 usually have them as well.