Answer:
I am an animal and a hair product
Squeeze me and I cry tears as red as my flesh but my heart is made of stone.
The more you take, the more you leave behind. What are they?
Deep, deep, do they go. Spreading out as they go. Never needing any air. They are sometimes as fine as hair.
I am not alive but I can die. What am I?
Some women don't like to be called that, yet every woman wants one. It cries and makes a lot of noise, but smiles and gurgles when you give it toys.