q1: treat Mr. Mustache as a generic cube. When you attach the cubes to the (cube)collider, also attach that pawn, making it a part of the slice. Then detach after the animation.
q2: what are the allowed moves? Do you want to move around on a single surface? Hop from one surface to a neighbouring one? Move all the way around as if was a globe? Is it one tile at a time, straight line, L-shape? A zig-zag, teleport, random? All above? 
If I’m not careful, can I fall into the nothingness below?
And most importantly, can he shave?
edit: I like how removing a couple of cubes from a single slice makes the whole thing an order of magnitude more complicated all of a sudden. On the other hand, this is coming from someone who has never held a Rubik’s cube…