Oooo,great question. For further reading on 'first contact' between the Europeans and Yuan China,please take a look here ,where you might read up on the papal diplomats between Yuan and Europe,like Giovanni da Montecorvino or John of Marignola. Eh,I can't say much about Yuan China so perhaps someone else can help out with that but I can certainly try my hand at the other two questions!

TLDR : To my knowledge,I don't think anyone knows or can definitively prove/find a Chinese family of European descent or a single European living in China from this period.

A (My) partial understanding of the Christian presence in Yuan China

To talk about European presence in China is also to talk about Christianity in China and the two are often used interchangeably. Native converts were practicing a foreign religion and hence were,themselves,"foreign" in nature,especially during Ming China. Hence,for this question,I'll talk about how Christians "disappeared" in the time from the 13th to the 16th century. I don't know of a single European living in China mentioned to be explicitly foreign in the Ming Annals though my guess is that they'd be either expelled or killed as per Ming ideals at the time. It could be likely that a few Europeans might've survived purges by fleeing to remote countrysides and then starting families but I don't know of a record which mentions this. Maybe someone else can help ! Most of the stuff that remains are very vague references to Christianity practiced by (probably) local adherents.

Anyways,as regards your first question,in general,yes. Giovanni spent about 30 years in China,during which he built churches in Khanbaliq (Dadu/Beijing) and brought together many converts,schooled them in Latin and Greek and translated the New Testament and its psalms into Mongolian Uyghur,before dying sometime in 1328. By all accounts,mainly the letters he wrote back home from 1305-06, it was a pretty successful mission,despite his claim that his work was being opposed by Nestorians in the Khan's court. His subsequent colleagues, such as the German Franciscan, Arnold of Cologne, were also sent to Khanbaliq to assist him though not all managed to survive the journey. At the time of Giovanni's death,there were three mission stations in Khanbaliq and one in Amoy harbour,opposite Taiwan (modern day Xiamen),undoubtedly as a means to welcome any future colleagues.

Christian influence was also strong up till the last years of the Yuan (1350s-1368) as Giovanni de' Marignolli,an explorer and an ambassador for Pope Benedict XII reports in his Cronica Boemorum the presence of "certain Christian Alanic nobles of high influence" in the Khan's court during the mission of 1338 and that the Khan Toghon Temür ie Emperor Huizong had specifically requested for the Papacy to send a replacement for Giovanni who had died some 10 years ago. Marignolli was insistent that "Cathay" was not a lost cause for Christian evangelism,while believing that the Khan was partial to their efforts and pointed out that the peerage of "great baron" ( magni barones aspicientes solum ad personam regis) was only conferable to "Europeans" (which included Semuren and "Alans") in the Khan's court as proof. During the fall of the Yuan,however,which lasted some 15-20 years,most of these records went up in smoke. We don't really know where the Europeans or Alans went but the likely scenario was that they were expelled or killed,leaving the native Christian community to flounder for itself.

But then everything changed when the Mingbenders attacked.

As you rightly mentioned,the early Ming were harsher towards "Mongol influences" which included an expulsion of Christians,a disdain towards Islam and everything not "traditionally Chinese",culminating in the "disappearance" of the papal mission of Guillaume du Pré and his 50 Franciscans in 1370 but we don't actually know whether there was internal persecution within Ming China where Christians and/or Europeans were actively persecuted but it seems very likely considering how these Ming Christians behaved. (The Hongwu emperor,Zhu Yuanzhang,1366-1398, was particularly predisposed to Islam and allowed its continued practice)

Matteo Ricci arrived in Macau,China in 1582,fresh off the boat as it was the sole Portuguese stronghold in the Far East for the last 50 years and the lynch-pin of any and all Portuguese,and by definition European,activity. What did he see ? A few locals had converted to Christianity and became translators and baggage clerks for the Jesuits but the Christian mission in China was, by all accounts,dead. Christianity had to grow all-anew from the time of Marco Polo and Montecorvino and Ricci took to it with great zeal. He was probably the second major missionary who seriously attempted to study the Chinese language,with help from his mentor Michele Ruggieri and realized that to mold China towards a Christian faith,he first had to mold Christianity to a Chinese palate. Thus began his long odyssey,from city to city, where he made friends with the local literati,most famously Wang Pan and Chu Tai Su,until he finally arranged an audience with the Wanli emperor in 1601.

Throughout this journey,but most especially after his meeting with the court in Beijing,Ricci was determined to find evidence of earlier Christianity but found his efforts futile until his renown and fame grew all over China as the great astronomer and mapmaker. Once there,he finally had some luck. In 1605,he was visited by a Kaifeng Jew named Ai Tian,who had already passed the civil service examination and was in Beijing to secure a position in the Imperial Court. He had heard that there were a group of foreigners in town who were monotheists but were not Muslims. Assuming that Ricci was a Jew himself,he arranged to have a meeting with Ricci. This was how Ricci first learnt of the Kaifeng Jews and hence hastily inquired as to the whereabouts of the "old adherents of the Christian faith". Ai Tian answered that they were mostly in Kaifeng (the ones in Kaifeng were actually Jews) or scattered all over China and were known as the "hui of the tenth number" as the Chinese word for ten "十" looked like a cross.

He mentioned that these "huis" made the sigil of the cross before they ate,carried a small black cross with them and had a reputation as great warriors but was unable to provide greater details. Fear of persecution had forced them to either abandon the faith,practice it in complete secrecy or simply adopt Islam. Ricci was gladdened to learn that such communities existed,at least in theory,but he never found any for the remainder of his life probably because these communities were too afraid to reach out to him or simply had forgotten the faith. Nevertheless,this was the confirmation he needed that the country he had found himself in (Sinae) was the same as the Cathay of yore (there were a few circles,most notably his boss Superior General Acquaviva back in Portugal who believed that they were 2 different states) and hastily wrote a letter to Acquaviva in July 26,1605 to restate this thesis and urged the Jesuits to commit more resources to his cause to "restore the large number of Christians who had once prospered in the days of Marco Polo".

Soooo what happened to these Christians from 1368-1605?

I'm sorry to say this after all that preamble,but no one really knows. We also don't know whether these dwindling Christian communities had at least some European genetics mixed in its ancestry dating from Yuan days. The late Bernward Willeke actually did some work on this back in the 80s. He found some evidence, namely letters and genealogies, dating to the 1500s that implies they and their descendants practiced Latin Christianity long before Ricci's arrival. Interestingly,none of these letters were written from major cities or coastal ones,indicating that these Christians had moved away from the metropolises to quiet hamlets and villages to practice in peace to escape persecution and harm. Hence,he speculated that Christianity might actually have persisted well into the 1500s despite the authorities,albeit suffering from attrition and lack of renewal. In 1543,however,Muslim unrest in the western provinces (Gansu and Xi'an) coupled with the complete withdrawal of the Jiajing emperor from all political affairs the previous year,leaving power in the hands of a conservative court, resulted in another general persecution of all "foreign based religions." (Cept Buddhism cuz double standards),resulting in all "foreign influence" being open anathema till the end of the 16th century.

Resources

Vladimír Liščák.The Christian Nobles at the Court of Great Khan, as Described in Medieval European Sources,2017

Bernward Willeke. Did Catholicism in the Yuan Dynasty Survive until the Present? Tripod 47, 1988

Michael Fontana's Matteo Ricci: A Jesuit in the Ming Court