Does Monster Power affect Staff of Herding component drop rates?
I know that Monster Power affects drop rates of a lot of things including the keys, but the pieces of the Staff of Herding pre-dated the MP system.
Does a higher MP affect the chance of SoH pieces dropping (specifically the plans)?
Since the plans are legendary items additional magic find can help them appear, from that point of view you can get 475% MF with MP10 and hit him in Normal or Nightmare.
Seeing as they can drop on any difficulty level, farming Izual/Diablo in normal or nightmare is a pretty straight forward way to get the plans.
With that said the plans are in a different bucket and appear far more often than other legendary items, so quantity of kills is ultimately what will get you the pop fastest.
All the other components can be farmed from the nearest waypoint in normal for extreme speed.