Mongolia is in a tough neighborhood. My understanding is that it was ruled by China’s Qing dynasty (after Mongols had themselves conquered China a few dynasties back), but became independent in the early 1920s, as China was mired in the warlord era, but only under Soviet pressure, and it was communist. Under continued Soviet and later Russian pressure, unlike other Qing territory ceded to foreigners (like Hong Kong or Taiwan or, to some extent, Tibet), China has not claimed Mongolia.
Yet.
Correct, as per Chinese propaganda, both communist Chinese and Taiwanese, Mongolia is a part of China.