How is it chauvinistic to allude to a shared heritage? Tibetan is pretty much the only major linguistic relative of Chinese, both being in the Sino-Tibetan language family. Other East Asian languages are in different language families and only share vocabulary with Chinese because of cultural influence in the past.

The majority of people who are ethnically Mongolian live in China, many of them being proudly Chinese and certainly wouldn't like someone like you insinuating that they are somehow less Chinese than Han people. I don't go around saying that Jewish people aren't German or Black people aren't American and I wish Westerners would afford our minorities the same respect.

Does that mean China should take over Japan too?

Nice strawman. Nobody in the screenshots is claiming that China should "take over Japan". Also there is no shared origin between the Japanese and Han people. There's only a small minority of people in Japan who believe that. You also assume that it is Han people who are going out conquering other peoples on the pretense of a shared origin when it's the other way around. Groups like the Mongols and Manchus were the ones who conquered Han people and then later decided to promote a shared national identity. For example emperor Kangxi mentioned in the OP claiming that his people the Manchus also being descendents of the Yellow Emperor.

This whole post is suspicious. I have no idea whoever translated this post is referring to when he uses weird ass terms like "Chinese Orthodoxy".