Firstly, I'd hope modern bottom brackets are greased.... The term you're searching for is cup-and-cone, as opposed to the cartridge bottom brackets more commonly seen today.

Almost all bottom brackets have the fixed cup on the right-hand side. However, I think Trek made a few mountain bikes which came with a BB that had the fixed cup on the non-drive side, and a left-hand threaded lockring and adjustable cup on the drive side. Quite why, I'm not sure.

I also put an italian BB back together with the adjustable cup on the drive side once. The damn thing came loose within about 5 miles, since BBs with right-hand threads on both sides need the drive side cup to be extremely tight to stop it coming loose by precession, and I evidently couldn't torque the lockring down that hard.