Considering this statement, "suppose we use an output cell size of 0.5 kilometers, and we see that one cell receives a value of 16 crimes per square kilometer. Since the area of the cell is 0.25 square kilometers (0.5 km * 0.5 km = 0.25 km^{2}), we can multiply the density by the area to get an expected count of 4 crimes (16 crimes/km^{2 } * 0.25 km^{2} = 4 crimes)...."

....does the square root of Cell Size (Kernel Density tool) equal the area of each cell (I'm using square miles for Area Units)?

Cell size does not describe the area of the cell but ... for square cells ... the length of its sides, hence, cell area = cell size squared