The 7x57 ( aka 7mm Mauser, aka .275 Rigby) is an awesome cartridge. Introduced in 1892, it is the 3rd ever rimless cartridge, and something like the 5th or 6th ever designed for smokeless powder, yet 122 years later, with the right load in the right rifle, it can still nip at the heels of the .270. They really got it right when they designed that cartridge.
The original load was a 173 gr. bullet at a reputed 2300 fps. Some years later, a 140 gr load was introduced, at around 2800 fps. Handloading can improve those figures significantly in the case of the 173-175 gr, and to a lesser degree for the 140. American factory ammo is loaded down to lower pressures in deference to older, weaker firearms. European loads are quite a bit hotter. The best place to get ammo is to roll your own, and you will need to take up that hobby to get the most out of it.
I've had a fascination/love affair with the 7mm for many years, probably as a result of reading Karamojo Bell and Jim Corbett, and to a lesser extent Jack O'Connor. I eventually started a custom project on a Mauser action, and with a bit of luck and nagging, it should finally be complete for next fall. This was the end result of a lot of thinking of what the perfect deer rifle would look like. With the scope removed, it comes out at a bit over 7 lbs., and is as handy as most .30-30 levers for still hunting, tracking, dogging, but with a lot more punch. The scope adds a pound and a bit, and lets you reach out to 300 yds and beyond if you have the skill.
I would take the 7x57 over the .308 any day. Velocities may seem similar with equivalent bullet weights, but the 7mm has much better sectional density and ballistic coefficient for any given weight.
For North American game, I would be comfortable taking anything up to the big bears ( brown, grizzly, polar) and bison. I would prefer a little more gun for elk, but I've no doubt the 7mm will get the job done if I do my part.