No- the 7x57 was developed separately, in 1892 by Paul Mauser. The 8mm Mauser (actually the 7.92 Mauser, but every calls it the 8mm) was developed by a government board.
Depends on your definition of better. All 3 are capable of taking any game found in North America. The 8mm Mauser carries a little more energy than the .308. The .308 may have an edge in accuracy. The .270 is basically the 30-06 necked down to .27 caliber, shoots flatter, due to higher velocity. This is one of those Ford/ Chevy/ Dodge discussions.
Yes
The .270 is a necked down 30-06. .270 WSM is a necked down .300 WSM. The 270 WSM is faster than the .270, and has a shorter case.
The parent case of the .270 is the .30-06, necked down. The parent case of the .243 is the .308 Winchester, necked down. Aside from the larger diameter of the .270, the .270 has a longer case.
The Mauser serial number 2082 is not specific enough to definitively determine the year of manufacture, as Mauser produced many models over different periods. However, if this serial number refers to a specific model, such as the Mauser 98, further information about the specific markings or features would be needed to narrow down the production year. Generally, Mauser rifles from the early 20th century often have serial numbers in that range. For precise dating, consulting a collector's guide or an expert in Mauser firearms would be advisable.
Send me photos of the rifle and of the markings, and I should be able to narrow it down for you. The email address is in my bio page.
The "parent" cartridge was the .250 Savage, necked down to .22 caliber.
The .243. It's basically a necked down .308 Winchester.
no- very much a center fire cartridge. It is the .308 cartridge necked down to a smaller (.243) bullet.
The .357 Sig is a 9mm projectile, loaded into a necked down .40 S&W casing.
There are several rifles, both military and commercial, which have been made for this cartridge. Try to narrow down your criteria to something reasonable.
I believe the smallest bullet in terms of diameter is the .17 HMR. It is a .22 casing that is necked down to accept a .17 caliber bullet.