Some nebulous thoughts before I rush out tonight:
Certain native Taiwanese do consider themselves different from Mainland Chinese people, even those that setlled on the island over fifty years ago. There is a clear divide between the two groups in Taiwan, which makes it impossible to group "republican" and Taiwan together. It is a touchy and complex subject that rages daily in the streets. My personal feeling, my family being of mainland origins and belonging to the KMT, having grown up largely in Taiwan and maintained residences in various parts of the world where Chinese communities exist, including China, is that while there certainly are differences, the similarities overwhelm them. Different dialects are spoken, different cuisines enjoyed, but these constitute regional subcultures. Urbanites and rural people are different everywhere but would normally not constitute disparate cultures either.
To your original point, I believe, in fact, that Taiwanese people in general are more likely not to claim cultural superiority over the Koreans and Japanese than the Chinese, not the other way around. Partly because some do not feel kinship with the imperial court (on the mainland), though textbooks have and still adopt a Sinocentric view of the world. They are changing towards a more Taiwan-centric stance in some ways. But because of the colonial history of Japan, the country being the first to surge out of the region to be an international power (economically), and its rise of cultural influence (which I think is most important), Taiwanese have undergone a sustained Japanese fever, and recently a Korean fever. This is mostly among the younger generations who do not remember nor care about WWII. Also, Japan has an ongoing conflict with China, too, and as a result, doesn't treat Taiwan poorly these days. And China plays up every risible Japanese incident.
As for Korea, that they have never really gone out of their peninsula (though for a long time they were the dominant sea traders in North Asia) to war and to occupy others, they do not engender the same kind of hate as the Japanese. Not despicable, violent brutes that is. North Korea being a playground of some sorts for the Sino-Soviet drama for a long time, adn being Communists, of course, they don't figure into the CPC's and people's despised races. I do not think that feelings towards Korea and the people tend towards the similar kind of hate, but they are surely not well liked. I would say that among people I've met and know, there is more uniform displeasure with them than with the Japanese, who provoke a wide range of emotional responses, trending positive. THe main points of contention with the Koreans are their insularity, arrogance, and nationalism, all of which are interrelated. I think that combination doesn't win many friends anywhere. EG I suppose the USA doesn't really have the first or third, though it could be argued otherwise, but even the perception of the second has caused a great deal of problems.
The relationship between China, Korea, and Japan is quite interestnig. For quite some time, much between Japan and China went through Korea.
Also, you are somewhat right abuot the US-KMT-CPC dynamic. Well before 1971, when the PRC took the ROC's seat in the UN, the ruling party in Taiwan was dissuaded from launching a strike to reclaim the mainland by various nations which wanted to avoid a hot war. This wasn't support for the PRC, but a pragmatic stance - te mindset continued and still does. Support for Taiwan has manifested itself in many other ways, but no one wants a destabilising situation to erupt in the Taiwan strait.