To take an amateur crack at your discussion questions:
The shape and methods of any continued conflict after the bulk of the exchange is probably the toughest question to answer. I can't imagine any national or alliance-wide command and control structure that wouldn't be grossly disrupted by a full-scale nuclear exchange, so what I would anticipate is that units still organized and extant enough to accomplish their missions would pursue those missions, then come to a halt awaiting further orders. Any Corps or higher level headquarters operating in South America and Africa might remain relatively unscathed compared to the devastation that would likely visit SACEUR and their Soviet equivalents.
In Europe, most combat units would be severely attrited both by conventional losses and by the wide-scale application of tactical nukes. The loss of national and regional command structures would probably cause everything to grind to a halt from lack of logistics and lack of coordination. Surviving command structures would probably be Divisional or Corps-level at best, which would not provide enough logistical tail to sustain operations for long.
Amusingly, the 2004 Battlestar Galactica is probably one of the best pop-culture examples of a transition from pre-nuke to post-nuke command structures. In similar fashion, any naval battlegroup would probably be the most intact military unit so long as they didn't end up on the receiving end of nuclear torpedoes or missiles.
National recovery of direct targets would be shaky at the absolute vomiting-rainbows best, and largely determined by how horrific the fallout and resulting nuclear winter became. Major metro areas, anything near a military or government facility, and any place downwind of those areas (especially nuclear missile bases) would be no-go areas for decades if not centuries. Anything downwind of North Dakota, for instance, could be completely written off.
The greatest loss would be communications - every major data transfer point between continents will have been erased from the map, and much of the orbital space around Earth would be a graveyard of fried satellites.
Beyond communications, survivors in developed nations would have to wildly adjust their lifestyles to accommodate devastated intra- and international trade. Between direct and near-direct nuclear hits on logistical hubs, widespread EMP effects on civilian (non-hardened) infrastructure, and radioactivity scaring everyone away from travel, transportation of materials would come to a virtual standstill. Very few port cities in the northern hemisphere will have survived, much less remained operational. Any power plants that survived and were still able to generate power wouldn't have anywhere to send the power due to the particular vulnerability of long-distance electrical lines to EMP. This lack of power means any intact gas or oil pipelines and processing facilities wouldn't be operational, and lack of diesel and gasoline would sideline any trucking that wasn't already fried by EMP. Again, regional networks might survive, with chances rising the further away you get from population centers and military bases.
The end result of this total logistical disruption is mass starvation and epidemic events in the largest pockets of survivors. Most major population centers are not self-sufficient in food, reliant on distant farms trucking produce and meat into local markets. Surviving medical facilities and personnel, meanwhile, emerge from under the initial wave of casualties only to find no resupply for all the materials and drugs and equipment they used, no water due to the loss of the national power grid, all followed swiftly by no power as their emergency generators run out of gas. People die in the tens of thousands of hunger or sickness, causing a secondary disaster as if the original nuclear devastation wasn't already enough.
Non-primary targets would still lose horrendously. Some countries that on their face don't seem like logical targets would still be directly hit. Spain, for instance, wouldn't be a primary target for Soviet nukes, but Naval Station Rota and Gibraltar would be, so Spain takes multiple direct hits. Anyone north of the tropics would be utterly screwed by the combination of widespread fallout and nuclear winter - the question isn't whether they would survive, but how long it would take for their populations to drop below non-viable numbers. Even the southern hemisphere would be badly scarred - Indonesia, for example, would suffer from the fallout of a nuked Singapore and probably a light dusting of Diego Garcia (not fallout from devices initiated over Diego Garcia, but bits of the atoll itself). Even for nations not directly hit by nukes, it's not a question of "Who wins," but more of "Who loses the least."
The ones who lose the least would probably be nations like Brazil, Argentina, Chile, and possibly some of the central African nations - these nations would be developed enough to survive the fallout and adapt readily to a world where American, European and Russian logistics have ceased to exist, but not developed enough or connected to the old world enough to eat a nuke like South Africa, Australia, etc. After the initial wave of deaths and the secondary wave of famine and sickness, China and India might have enough survivors, enough local organization, and enough local subsistence ability to begin rebuilding the places that are safe enough to rebuild. Who knows - some of their nukes and national command may have survived to start a new geopolitical cold war across the Himalayas!