In terms of cartridges that are tough to beat, I'd say 6mm Dasher (7 or 8 twist), 6.5mm Creedmoor, 7mm-08 and 7WSM/SAUM are examples of very well designed cartridges for hunting in North America. Short, efficient, of various power levels, in the right high sectional density metric calibers, with no major design flaws. The Dasher and 7WSM/SAUM suffer from lack of support though making .243 and 7mmRM more practical choices even though the cartridge design is worse (quite a bit worse for 7RM).
If dangerous game is in play, I would point to .404 Jeffery and .450/400 Nitro Express. There really are no efficient dangerous game calibers (since an efficient shoulder would make the case absurdly wide) so it's just a matter of getting sufficient sectional density and enough diameter while keeping recoil low enough to allow reasonable follow-up shots. Those two have all three properties.
For lever gun dangerous game cartridges, I would look to the .45-90 and .50-110. Not much design to be done really - just needs to fit in an 1886 and hold a big honkin' bullet in a common caliber.
