Can there be a run where you have to go out of your way and kill every enemy before beating the game? Oviously all enemies would have to be mapped out, but if that happened would it be possible?

Hmm, would that include controlling all moe-eyes, cacti etc. into suicide? If so, it could become dull quickly

A better idea might be to do that as a one level thing instead of whole game.

How would you verify that a run is valid?

Panda I think you're right, each world would have to be it's own thing. As for verification of a run would have to pretty much be a check list of each enemy

Is it literally killing all of the goombas and piranhas etc. Or just one of each type?