The glycocalyx is often referred to as the "cell coat," which is somewhat fuzzy and sticky with numerous cholesterol chains sticking out from the surface of the cell membrane.

The statement above is FALSE.
The glycocalyx refers to a glycoprotein covering which surround the cell membrane of some bacteria, the epithelia cells and some other cells. In animals, the glycocalyx in the epithelia cells is in form of fuzzy like coat which cover the external surface of  their plasma membranes.
The glycocalyx in bacteria help them to resist engulfment by white blood cells or other predators.
The glycocalyx is made up of protein and sugar and not cholesterol.