Where to begin?
Lets start with your presumption that the PRC gets 'a few' nukes off. The actual number will likely be more like a few dozen or so, though most of them will be very, very short ranged. No more Vladivastok, Irkutsk, Khabarovsk, etc. Possibly no more Hanoi/Haiphong (though why the Chinese will waste precious nukes on distinctly secondary targets is beyond me..), and likely several Indian cities will be slagged if they are foolish enough to enter into a war (they wouldn't by the way). Major Soviet airfield complexes will be taken out as well, and a few choice chokepoints on the TSR will be glowing in the dark for years to come. The PRC had a few IRBMs/MRBMs as well, so there is a decent chance that they will lob a few at Soviet population centers in central asia and the Urals region, possibly even one or two for the big targets of Moscow/Leningrad/Kharkov/etc. Tell you what, we will pretend that the largely worthless soviet ABM system around Moscow stops those, for if it doesn't, this whole thing ends much more quickly.
Once this happens, the entire soviet logistics network collapses, and their army runs out of supplies and dies within a week or two. POL and ammo would be the biggest problem (Soviet divisions carried only a weeks worth of each even in assualt load-outs) The Far East Military district probably has some stockpiles (lets pretend that the PRC is foolish enough to ignore ALL of them, as well as the distriubtion centers, which are large, easily attacked, and cannot be hardened by their very nature), so lets give them USSR about 4 weeks before their troops are literally eating their belts and throwing rocks at the PLA. The entire logistics network runs directly along the TSR, even the major airfields, so only a few bombs are necessary to do the job.
Of course all of this is going to look like paradise compared to what happens to the Chinese. Soviet strikes against airfields, missile launch sites, troop concentrations (harder than it sounds, the PRC is mostly light infantry, and they dont' bunch up much), and of course population centers (what they heck, they might as well go in for genocide while they commit suicide) are going to kill a VERY large number of people (if both sides do what we now know their pre-war planning involved, the figure of 100 million is likely to be low by at least half, maybe more, and that counts only prompt deaths, disease, starvation, social collapse, etc, could easily double or triple that), and destroy a large fraction of the infrastructure.
My point here is that any use of nuclear weapons will be highly problematic for both sides, but primarily for the USSR. China has a LOT of people, and it has enduring horrible suffering and loss in the past. Circa the Carter era (a good time for this...after all, the rest of the world was miserable anyway, why not them...), the PRC was far, far less developed than the Soviet Union, and thus had less to lose. To be crude, they could be bombed back to the stone age, but it wouldn't be a long trip. The Soviets, on the other hand, couldn't endure the kind of losses suffered in WWII (whether they beat the PRC or not, particularly in a war that THEY STARTED) again without a serious threat to their govt. An attack on the PRC, particularly a nuclear one, would almost certainly invite nuclear retaliation, retaliation that would be far, far worse for the Soviets than it would be for the PRC. Now I can imagine some scenarios when a war might start, but I can imagine absolutely none (post 1970 or so) where the Soviets would use nukes unless the PRC did first.
Lets move on... Why no mention of gas? The soviets had lots of it, their troops were well-trained in its use, and the PRC had no effective defense against it. Light infantry (the bulk of the PRC) is most vulnerable to chemicals, and the terrain is almost perfect for it. Gas makes a wonderful area denial weapon as well, which is going to be important to the Soviets, who are advancing significant distances through highly hostile terrain...
Now, for fantasy, one cannot beat your suggestion that India and Vietnam would join in this insanity. The Indians had their heads handed to them in 1962, and until the 1990s, their foreign policy was largely centered upon not creating waves with the PRC. Why would they engage in a war that would net them almost nothing (Tibet?...risk nuclear retailiation over Tibet?), and would cost them a great deal. Not to mention that they have a far stronger (and more profitable) role as a mediator when both sides tire of this silliness if they simply stay out and wait. Nobody but an idiot would fail to recognize that even the 'winner' of this dust-up is going to be in really bad shape. So if the Indians stay out and wait, they can pick up some nice tidbits for free (or nearly so) when the PRC is in no position to complain. If they get involved, they could lose a city or two. Which one do you think that these folks are going to choose?
Vietnam is more problematic, but since they would be attacking a force that outnumbers them roughly 3:1, and is deeply entrenched (and lets remember, the PRC won't be advancing INTO Vietnam, so the tactics that worked so well during the last PRC incursion won't pay off this time), not to mention nuclear-armed. Once again, what does Vietnam get out of this? <crickets chirping>....anyone?....Bueller?....
All of this discussion of Eastern Europe...do you think that NATO is going to decide that this is Der Tag, and go to snatch up the Warsaw Pact? I am sure that the Brits, Italians, etc (not to mention the French) will just love that idea, especially since it almost inevitably leads to German reunification. Want to piss of EVERYONE, just go for that. And the Eastern Europeans, do you think that they are going to cooperate? A few protests, certainly, maybe even some serious rioting. But remember that the bulk of the Soviet military (including the overwhelming fraction of their Category A formations) would still be in Central Europe, and I rather doubt that anyone is going to play games with them there.
Lets return to the far east. I like your war aims, they are limited, make sense, and are achievable. Best guess is that the Soviets would use Harbin as a target for their primary axis of advance, with secondary strikes enveloping it from both sides of the Manchurian 'finger'. Given the terrain and the nature of the opposition, they will bet on a 30-45 day campaign, with a contingency plan for 60 days. Lets assume that they can stockpile 120 days high tempo supplies (tough to do, but possible), and not give away their mobilization until 10 days before the head out.
Barring any real surprises, this thing is doable, though very, very risky. Remember that the Soviets didn't have any real deepstrike capability at this time (I am working with 1979 as my target date, earlier - say 1972 or so - things go nuclear very quickly, later - say 1984 or so - the PRC is much stronger), so the PLAA will still be in operation and we should expect some truly awesome dogfighting as two huge incompetent airforces tear each other apart. Soviet armored formations should be able to penetrate very deeply without much difficulty, and assuming extensive use of gas, PLA casualties should be very high. I agree that eventually things will bog down, but not before the Soviets roll into Harbin, snipping off Manchuria and switching over to a garrison mode. The Chinese might wish to continue the war in low-intensity mode (guerillas, terror strikes, whathaveyou), but the real fighting would be over in about 5-6 weeks.
The aftermath would be horrible for both sides, which is why this would only happen if the Soviets were convinced something worse was in the offing. That scenario I leave as an exercise for the alert reader...