GWU has a great national security document archive with related content.

I can't speak directly to your question about the SEATO allies, but here is the summary of a 1963 NSC war game titled '"The Management and Termination of War With the Soviet Union."

At the essential breaking points, the war-game president has the choice of striking Communist China along with planned strikes on the Soviet Union. Both times, he chooses to include Communist China in the strikes. The summary concludes:

It should be noted, finally, that the decision-making process took place in haste and under unimaginable psychological stress, and that emotional factors would have pushed the decision in the direction of a maximum retaliatory attack. It is unlikely that under such conditions the decision would be reached strictly on the basis of ad hoc gain/loss calculations.

Further along in the same pdf is a 'War in the Far East' scenario specifically focused on Communist China, which may be of interest to you. Included is one of the most striking quotes from the whole affair:

Also the US did not want to give the appearance that it was always easy to decide to use nuclear weapons against Orientals.

Many of these places had US airbases on them that would serve as launching points for either nuclear attacks (e.g. Okinawa) or, more commonly, as launching points for refueling operations (so that bombers which took off from the continental US could get a fresh fuel load before trying to penetrate Soviet airspace). This is definitely the case in terms of the three countries you named, all of which based some US nuclear weapons (I don't know about the strategic/tactical mix) for much of the Cold War, and at least Japan also housed refueling bases. So they would be intimately involved in the attack by as a result of their political geography (hosting US bases), and probably targets for a counter-attack or pre-emptive attack.

SIOP-62 would have China treated like any communist country (bombed to bits) however as SIOP-62 was scrapped for being too inflexible there multiple options available depending on where or not China entered the war. After the Sino-soviet the chances that China would enter into the war.

As most of the bomber/missiles would fly over the article circle to hit Russia everywhere there was little reason to place nuclear assets in the east. Instead any fighting there would probably be focused around containing Russian naval forces around Vladivostok but given how easier it is to blockade that area the Russians didn't deploy any missile subs there.

That being said there would probably be strikes against US military bases in the region and early warning stations, either conventional or nuclear weapons.

As for American allies? Secure they're own territory (this of course would mean that they would be bought into the conflict by attacking soviet aircraft/ships attacking US bases on their territory) and prepare for nuclear war. There is also the possibility of the Korean war kicking off as well.

I can only really speak about Korea here, but my impression is that immediately following WWII and after Communist victory in China was more or less inevitable circa 1947, America's Asian strategy was to attempt to pull the PRC away from the Soviet Union and into either a neutral or US friendly political position. Some of the overtures made at that time included moving South Korea out of the US defense umbrella to give China "breathing room," so to speak.

Korea was seen more as a bargaining chip or a buffer zone than an asset. One must understand that at the time, Korea was among the poorest nations in the world, with ongoing insurgency against the US appointed government, only rudimentary infrastructure, and limited strategic value in a geopolitical climate focused largely on containing the Soviet Union's influence in Europe. The United States felt they could better serve their interests and protect the more valuable Japanese archipelago by either making friends with China or forcing them and the Soviet Union to fight all the way down a rugged, underdeveloped, notoriously harsh peninsula should war come.

Of course the Korean War changed removed any possibility of amicable relations with China for the next two decades, and afterwards Korea was seen as a more valuable military asset. Korean troops were even used in the Vietnam War where they proved themselves to be among the toughest and most ruthless fighters.