Several of your gut instincts are correct here. A great number of German soldiers did suffer from frostbite and exposure, and it did reduce the combat effectiveness of frontline formations. However, the Germans did undertake a variety of both official and ad-hoc policies to mitigate the problems of extreme cold. At all levels from army-level leadership down to individual squads, leaders encouraged their troops to "requisition" whatever resources were needed from the local population. This included food, peasant huts, dry firewood, and warm winter clothing - all without a care as to the suffering inflicted on the victims of this requisition. There are a multitude of documented cases, as well, of German soldiers exhibiting needless, excessive cruelty within this dynamic. For instance, a colonel in the 20th Panzer Division billeted himself and a subordinate in a peasant family's hut sometime in October 1941. The colonel, after having spent much of the night drinking, announced that "‘I can’t stand the sight of these peasants’ faces!" and without warning shot the father in the face; a short time later, he shot the wife and children because he was sick of listening to them cry. For this crime, he was relieved of command, but suffered no further punishment. In other cases, the murder of peasants was less direct but no less the fault of the German soldiers who occupied their huts. A Wehrmacht soldier named Henry Metelmann recalled in his diary that

Our orders were to occupy one cottage per crew, and to throw the peasants out. When we entered ‘ours’, a woman and her three young children were sitting around the table by the window, obviously having just finished a meal. She was clearly frightened of us, and I could see that her hands were shaking, while the kids stayed in their seats and looked at us with large, non-understanding eyes. Our Sergeant came straight to the point: ‘Raus!’ [Out!] and pointed to the door. When the mother started to remonstrate and her children to cry, he repeated ‘Raus!’, opened the door and waved his hand towards the outside in a manner which could not be mistaken anywhere … Outside it was bitterly cold … I watched them through the small window standing by their bundles in the snow, looking helplessly in all directions, not knowing what to do … When I looked back a little later, they were gone; I did not want to think about it anymore.

In addition to stealing food, shelter, and warm clothing from the peasantry, Soviet POWs and dead soldiers were often looted of their warm winter jackets and socks. It was not uncommon to see entire columns of POWs stripped down to their underwear, shuffling along like penguins huddled together, while their captors wrapped themselves in Soviet greatcoats. Callousness towards the population was inextricable, then, from the Wehrmacht's attempts to cope with the harsh winter, and from the war's conduct in general, and this must not be lost sight of.

See David Stahel's Operation Typhoon: Hitler's March on Moscow, October 1941 for more on the conditions during the winter drive on Moscow, and then Omer Bartov's The Eastern Front 1941-1945: German Troops and the Barbarisation of Warfare more generally.