To answer Yahtzee's nitpick.
The reason some doors do not open, you'll notice it's only in the "Time and Place" world, is because you're approaching the door from the right (walking left towards it).
If you recall, walking left reverses the flow of time. For the same reason that jumping left on top of a goomba doesn't kill it, walking left with a key into a door doesn't open it. The game's rules of cause and effect require that time be flowing forward.
I agree that this particular mechanic was a little tedious and ultimately added very little to the gameplay, but it was far from an arbitrary mechanic, it served to illustrate one of the consequences of that particular time manipulation theme.