Japan, south Korea, and Taiwan all have extremely unfavorable views towards China, like upwards of 70%. I don't think there is really any chance of them taking China's side in the foreseeable future.
Nobody doubts that the PLARF could annihilate any and all neighboring countries with nukes, and the underground great wall and city subways could hold hundreds of thousands of Chinese people underground. But any MAD scenario between China and its neighbors is still a huge loss. There is no nuclear war scenario where anyone ends up better off. In particular, Taiwan having a sizeable nuclear arsenal with launch on warning and second strike capability would not be something China could just overlook in military planning.
The concept of the American nuclear umbrella lacks credibility as China expands its arsenal, the thinking is of course that American presidents would not be willing to risk retaliation even if China nukes Japan or Taiwan first. By keeping Japan, Taiwan and Korea unarmed with nukes, the USA has created its own strategic vulnerability if it intends to defend them. Conversely, by arming them with nukes, the US creates a major dilemma for China.
Clockwork, it may be the case that European countries which lean pro-America (and many other countries) may be willing to look the other way if the USA proliferates nukes to its east Asian allies. But China giving ICBMs to actual jihaddist groups pretty much guarantees they would be used immediately. There is also the difficulty of getting land based large TELs loaded with ICBMs delivered to the groups you listed. It's possible, but I think the USA would be able to get away with proliferation among allies (by that I mean a lack of economic consequences) while China could not arm any of the groups you mentioned without blockades, sanctions, etc. being imposed by the majority of UN member nations. In my opinion, the USA has an asymmetrical advantage by doing this, while China does not.
