There's always a lot of "posturing" and "Sabre Rattling" that goes on in these situations, but pretty much everyone knows, there is a limit, and if you push beyond that, it will become your worst nightmare and the end of your country. I doubt that a nuclear exchange would happen between NATO and Russia because there is nothing to be gained. Everyone loses.
However, in the event that it did, the MAD concept is inaccurate. Many incoming missiles would be intercepted, but not all. Many would survive, but it would be a rough ride for sure. No one really knows if a nuclear winter would happen, as it would all depend on if the debris made it all the way into the upper levels of the atmosphere. If it didn't, the debris would become fallout. Severe radiation is generally limited to roughly a 50 mile radius of the blast (Depending on the megatons). Radioactive fallout would follow the prevailing winds and dissipate as it traveled.
Taking out military targets (Missile silos, etc...) would be the goal, not necessarily cities. That's why our nukes are placed away from populated areas, for the most part. There would be many challenges, food, water, gas, supplies, etc..., but many have prepared for worst case happenings, and they would make it through, and quite possibly many others. People can be very resilient.
Again, I seriously doubt this would happen over Ukraine, as it could mean Putin would sacrifice his entire nation just to gain some extra land, which he wouldn't end up with anyway. Even for him, it has to make sense on some level.