Stalin's Great Purge happened before USA recognition of USSR, not to mention back then UK was the superpower and it had already recognized USSR in 1924, 12 years before the Great Purge. USA didn't become a superpower until WWII. After the Great Purge, USSR and Nazi Germany had good relationship and signed an non-aggression pact and it was the Nazi that later started war first against USSR.
The USA supported the Nationalists in Chinese because KMT-controlled China was a pro-western dictatorship, which was always the case until the late 80s.
Nixon administration normalized relationship with China not because "to split the Communist world" because it was already split. China and USSR already had a bad relationship over nuclear bomb development, so it made sense that USA would cooperate with China to go against the anti-elite USSR.
Deng Xiaoping was the most pro-west ruler of China ever and his economic policy gave the elites a hope of China replacing USA as the world superpower, given the huge population of China. This is when the elite-controlled USA started to greenlight a series of economy policies that would help China become a superpower. China cooperated by keeping the labor cost down through dictatorship, therefore convincing the investors to transfer manufacturing to China.