Which part of human nail anatomy  is composed of hardened keratin, and is the most functional part of the natural nail unit?

The dead tissue adhered to the nail plate that can be cut or removed with a specialized product.

Which part of nail anatomy is living skin that supports the nail plate as it grows toward the free edge?

The visible part of this nail anatomy is the lunula, and it functions by forming and producing nail cells. Additionally, this part of the natural nail contains nerves, lymph and blood vessels. It is known as the bed epithelium.