At some point, Diablo 3 generates a map which may or may not have various random figures/events/dungeons. Does the Magic Find skill have anything to do with this? Specifically I'm wondering in regards to finding various parts of the Whimsyshire staff - should I load up on magic find gear before I go looking for things like black mushrooms and the mysterious cave?
No, Magic Find % multiplies the drop rate on the item its self. If an item has a .0005% chance to drop, and you have 100% MF then the item now has a .0010% chance to drop. So you may find it harder to get items, but it's not because the map or difficulty, Just plain old chance. An item always has a potential to drop.