cut a whimsical cake like the one in the tutorial on this site? If each layer is sitting inside the bottom layer, do you have to take the upper layer off before cutting it? If so, and its covered in buttercream, is there some way to do it without messing it up?

I think you should cut it in place because it is inside the next cake. That's my humble opinion!

I cut it while it's still stacked. I start cutting where the "tilt" is at it's lowest on the lower tier. Does that make sense? Once you get that first slice out, it's pretty easy to cut from there.