Texas Jack,
If you can post photos of the proofmarks and the extractor, we can tell you more.When marking the caliber, prior to 1939,it was not shown if it was rimmed or rimless. It is highly unlikely that it is rimless; but if so, it will have a spring loaded pin in the extractor.Also it would be unusual to have ejectors on a combination gun;but if it does,there would be two extractors, rather than one for both cases.If there is no #70 next to the 16, then it is "short chambered"(2 1/2"-2 9/16", instead of 2 3/4").For what it is worth,7mm is unusual in that the caliber is marked the same as the nominal caliber( pre 1939 proof law).Normally, the caliber would have been shown by the bore diameter, while the name of the cartridge would have been something else( ie an 8mm cartridge would have been marked 7.8 or 7.9), in the case of 7mm cartridges, the bore is 7mm ( groove/bullet dia. is 7.2mm).
The combination of 16 ga and 7x57R is very good and at one time was almost the "standard".New 7x57R ammo is primed with standard boxer primers,and is reloadable with standard bullets and normal dies. The only difference is the shellholder, which is avaliable.BTW,in my opinion the stock has been refinished, and likely repaired.
Mike