How do limb, down, naild and teeth, grow? What "pushes" them out?

New body parts grow by division of stem cells at the remnant. They divide, differentiate into the correct cells and grow larger. The bulk of cell dividing and growing larger provides the "pushing" out of new tissue.
Your body is other making new cell. In the case of bones and blood, the unknown stuff grows to take the place of the outdated. In the case of mane, skin, etc., the new stuff pushes the antiquated stuff out. That's why your nails, spine, etc. keep growing! I hope this help to answer your question.
the cell at their roots- for nails, tresses and teeth, though all stop growing after a while. for limb, its internal growth due to growth of bones and muscles within.
By their property.
genetic engineering is base on the idea of end cells which can be developed into any cell within the body;finding the mechanisms which will trick into responding as they would to their deep genetic instructions is the fundamental problem