If the underlying surface is, for example, dark soil, it will absorb most photons that reach it. Hence a thin snow cover will appear darker than a deep one. In answer to the last part of the question, the scattering properties of a substance depend not only upon the substance itself but also upon its microscopic shape even the arrangement of its molecules. The water in a clear lake transmits sunlight much better than snow. Pure water is both weakly scattering and weakly absorbing.
Extinction is minimal, and so, even at ten-foot depth, the lake bottom is well lit.
With so little light being scattered back out of the water, its surface appears dark. We have noted that ice absorbs light weakly. It also does so selectively. Ice has a broad minimum of absorption in the wavelength region corresponding roughly to blue light, and absorption increases sharply toward the red. The same is true for water.