Dan, if a passive pickups hums and buzzes, it's because it's not grounded properly, or it's a single coil picking up stray noises. In this case, though, neither apply because the Bare Knuckle 58 is not a single coil but a humbucker. Also, while I do love EMG actives, they're a whole different ballgame tonally than passives, and personally I wouldn't use one to replace the other if someone's a fan of passives.
However, you are very much correct that added treble and drive create noise. When you boost a certain frequency range, you also boost that freq range's noise floor. And drive pedals can often be pretty noisy, and the more you crank the drive, the noisier they will get. Beans' suggestion of a noise gate pedal to shut your signal down when not playing is the only real solution. He suggested the RockTron Hush, which I have and is quite a good pedal, but pretty much every pedal company makes a noise gate pedal.