#6 should be "protecting national interest". It wasn't US

intention to attack PRC at that point in time, but due to MacArthur's pride and general mistrust between PRC and US, it became necessary. The whole thing really was a setup by USSR to test PRC's committment to Communism (actually to Stalin). US failed through their clouded political ideology and PRC had no choice but to fight in Korea once US blocked Taiwan. Thank god that PRC won that war. If not, then all hope of China would be lost.

#13 PRC lacked power to break through US carrier at the time. Therefore can only fight in Koreas

#17 strategical decision. Using "volunteers" had avoided the necessity of declaring war. PRC wasn't strong enough to declare all out war on US nor US wished to declare total war on PRC. One part of both of their reasoning was due to the existence of USSR.