I am not certain that the game has gone to great lengths to model Russian weather patterns as opposed to the rest of the world. I certainly see no evidence of the Winters of '41 and '42 being unusually cold (even by Russian standards).
The aspect of Russian weather which I think has the greatest impact on play is the spring mud. It is quite bad to be attacking during that as you will typically lose lots of equipment to no avail. In SP one can get away with such waste, but it is still better to pay attention and avoid attacking during those weeks.
The game's treatment of weather particularly in Russia is one aspect where I feel there is lots of room for improvement. If the logistical model had a bit more depth to it tying winter into supply difficulties would be nice. Attrition while fighting and moving in winter is fine, but if supply is also bad that will make getting replacements to the front more difficult. As Winter drags on some of those units should start looking pretty ragged.
The Russian should absolutely not be exempt from the above effects but this is where their greater numbers can come into play. If they are conducting a very expensive Winter offensive they can afford to rotate troops away from the front to bring up relatively fresh ones. With the logistical situation mentioned above this should be able to force the Germans to give ground in some areas.
I would also like to see even the larger rivers freezing over. River lines are a huge strategic factor in the Russian war and many if not most of them did in fact freeze over.