We weighd our .22lr and sorted them according to weight for Biathalon, and there IS a difference, but thats the only example I know of where the whole cartridge was weighed.
As to a Mosin, give it a try when you buy it. Use the ammo you plan to hunt/shoot with.
I use Czeck light ball, it is VERY consistant, and if you go to the Hunting section, I posted under "Brownbears and Wolves" , you can see just how accurate an Issued M-39 variation of a Mosin is, with good milsurp cartridges.
As for tuning, I start with a good clean rifle, make sure my action is good, tighten the stock screws and shoot. Then I adjust my barrel bands with tightening or loosening, as that very often makes a big difference, clean and hone my trigger working surfaces, and if that doesent work, and teh barrel crown is good,I might shim the stock up a bit. Thats always been as far as I needed to go, never had to chop stocks or goof with any metal.
Anyway, there you go, comnsistant ammo is where its at, same weight and charge should get it where you want it every time.