The race would have been "on" between the Chinese and the Muslims for the conquest of the New World. My money would have been on the Muslims; unlike the Chinese, they had an imperative to go out and convert the heathen to Islam. Also, they were pretty good seafarers.
The Chinese, OTOH, thought -- with some justification -- that everyone else was some flavor of barbarian, and every other culture was completely inferior to theirs. There was some interest in trade, but only very limited interest in any kind of conquest or colonization.
(Note that NW Australia is far closer to China than to anywhere in Europe or North America, but the Chinese probably never got there and certainly didn't colonize.)
Depends on the state of Islam during the time period. Are we talking about a bunch of Wahabbists? Then you wouldn't see any kind of scientific advancements. The Chinese would have been more likely to pursue technology.