How did Mongol and Chinese cultures converge
Actually, they did not converge; in fact the Mongols went to great lengths to keep the two cultures separate. The Mongols considered the Chinese to be "mere cultivators" and prohibited intermarriage between the two groups. They even forbade the Chinese to learn the Mongol language. The Mongols used administrators from areas outside China such as Arabia and Persia; but never used the Chinese. So great was Mongol scorn of anything Chinese that there was discussion at one point of exterminating them as a race and using the land as pastureland. Ultimately, it was decided this was not feasible, and the best approach would be to extract as much as they could from the country. It was the Mongols who forced the Chinese people to shave their heads except for a queue, which was an identifying mark. Although the Mongols were tolerant of many religions and even allowed the construction of Christian churches and Buddhist temples; they were not so generous with the Confucian system. They dismantled the examination system used to train Confucian scholars and ended privileges to which those scholars had been entitled. They did not actively destroy Confucian teaching; but did all they could to strangle it.
Chinese culture did not return until the Mongols abandoned China and returned to the steppes.