In Animation Paper you don’t first select, then cut. It works in one go.
So you just picked what ever you lassoed up. That’s it. Now you can move your floating selection around as you please, change frame, scale/rotate, etc. Then you stamp it down by hitting Return/Enter or just click outside the selection.
When you did the Ctrl+X you cut the whole drawing (like from the X-sheet). You still saw your selection, but that was because you had it picked up.
Does that make sense?