In particular, if there had been no Mongol conquests under Temujin, Sung China would have continued as a large scale mercantile power with much maritime activity. It is unlikely that the Sung state would have sponsored colonization, but that isn't a requirement.
Yes, but you're ignoring the devastation that occurred more than a century before then, when the Jurchen took over North China. Although the Song still retained the south, they were severely impacted, and it would have taken them many decades, if not centuries, to recover significantly. A significant amount of maritime activity certainly existed before and after the invasion, but there was no particular impetus to form colonies overseas, due to the lack of pressures which caused the Europeans to look outward.
Although the (Khitan), Jurchen, and/or the Mongols might not have invaded China in another timeline, other nomadic groups would have probably taken their places, given the long-term trend IOTL. Not only would the invasions have caused the Song to invest more resources into repelling the incursions from the north, but the decrease in population would have also led to very limited population pressures, with more resources for each individual. The constant invasions from the Khitan, Jurchen and Mongols also did not cause the Chinese to look overseas in order to seek better conditions, so I don't see any way for chaotic conditions causing widespread emigration to occur either.
Areas like the west coast of the New World and Australasia were thinly populated (and vulnerable to Old World epidemic diseases). Even scattered trading outposts would be enough seed to grow into significant settler colonies, and set the ethnic character of these areas.
If the Song retained the southern areas, trade would have still been conducted on a widespread level, and although there might have been some limited explorations overseas, there would be no particular reason to form stable colonies, due to minimal population and economic pressures, after recovering from nomadic invasions to the north.