Is there a way to estimate contact patch size given tire size plus tire volume and rider weight?
Basically: weight/psi = square inches.

Has to be combined weight of rider and cycle. And it's approximate. And much easier with a unicyle: with a bicycle you need to know weight distribution, which can change during a ride.

Minor things that make it only an estimate:


Compression from weight increases psi, making contact patch smaller than estimate. 
thickness of material could increase contact patch size. In fact, I think this math gives you the area of the part of the interior surface of the inner tube that's pressed against the part of the tire pressed against the ground
stiffness of tire materials could decrease contact patch size
tread features (knobs, grooves, etc) could do all sorts of things to contact patch size.
bumps, jumps, etc momentarily increase "weight" (downward force on wheel or upward force on contact patch), contact patch size and psi.


No idea about new fallen snow.