I am aware that there were quite a few missionaries living in China and other nations in the area, but were there any other Europeans living there? Like families? Traders? Courtiers? Criminals? That sort of thing. Were there people who either tried to make a better life there or even assimilated into the local culture/society? Specific examples would be appreciated if available.
You're certainly right that there were missionaries; Hong Xiuquan even declared himself the little brother of Jesus and lead the Taiping Rebellion during the Qing Dynasty. But that is, of course, during the colonial period, and beside the point.
To answer your question, during the Pax Mongolica (period of relative peace in the Mongol empire), many European missionaries and traders visited China and chose to stay. People remember Marco Polo, but he was one of thousands, and is only remembered because he published a book. There was an entire Genoese colony in Zaytun in the 14th century, mainly used for trading. Venetian traders, especially, would commonly go to China. This did not make them common in China, but it was not rare to see a European. A group of purportedly 30,000 people from modern-day Turkey made up the guard for the Yuan dynasty, according to Jean-Paul Roux in Histoire de l’Empire Mongol.
As far as "criminals" go, there aren't many documented instances of a criminal being sent to China, but William of Rubruck did describe a contingent of Teutonic prisoners in iron mines. Overall, Europeans were seen as barbaric by the Chinese, who believed their Emperor to be the chosen Son of Heaven and therefore above all other rulers, and thought that a similar hierarchy existed between the Chinese culture/people and all others.