Not the son, a junior party in the same government as the KMT, the CPC did not evolve from the KMT but was rather a partner to it.
Jiao zhi is just the general name of the area. moreover jiao zhi is literally several periods/governments/eras away from modern vietnam, you are cherry picking a singular period of history for your claims and not even a recent one. convenient for you to forget that even ming china owned half of vietnam and its half was Jiao Zhi providence of the Great Ming. Heck even the name of "Vietnam" was grant by the Qing emperor not to mention that north Vietnam had fully agreed to release any claims in the SCS, only to go back on its word once the country was unified. per history, which your claims are based on, china has a far better claim.
how is this illegal claiming? its just clarifying the maps now that the immediate danger of japan is gone given the absolute turmoil of the last 50 years, the then the owners of the various claimant nations, such as the master of Vietnam, the french, the owners of other near by areas such as the Americans and British, dutch etc not a single one disputed the KMT claims, hell the first KMT Garrison on Taiping island in the SCS got there on american boats, as in the USN literally shipped them there. and again, china has better claims than vietnam anyhow
the PRC did indeed claim those areas as it is the successor of the ROC. and through negotiations it worked out each of those border disputes, including the land border with vietnam, which the ROC on taiwan still claims. in fact the PRC has worked out the vast majority of its land disputes through negotiation, except prominently, with india, since india essentially refuses to negotiate and has wet dreams about a made up british-defined border that china had never agreed to and that even the british today agrees was illegal.