Answer:
My first is a heir; My second's a snare; my whole is the offspring of fancy; which I sent out of play upon Valentine's day As a token of love to my Nancy.
I mirror your actions.
Take off my skin I won't cry but you will.
Break me and I am better immediately set and harder to be broken again.
What goes up and down but does not move?
It has no top or bottom, but it can hold flesh, bones, and blood all at the same time.