China has the resources to do this ten times over. The only problem is that China lacks the motivation. So much so that this would be an idea that only a very, very bored mandarin could have dreamt up. I'd say a Chinese colony would probably be in Taiwan, the Philippines, Borneo (ie the Lanfang Republic), Malacca or Australia.
Certainly true. The best way of getting attention to the New World before Europeans do it is to have an expansionist Japan go up Hokkaido, the Kuriles, Kamchatka, and eventually Alaska. I'd think once it's realised that the coast of North America curves southwards and there's more temperate land (instead of just more frostbitten land where frostbitten barbarians live) they might have some idea.
It isn't like there's anything the barbarians on the other side of the Pacific have to offer, aside from maybe intriguing Chinese traditional medicine ingredients (once someone declares whatever American thing of value in Chinese traditional medicine). At least until you find gold and silver, which is decently inland and which the natives don't really have a clue of the scale.
If they go to nowadays Mexico, however, that's a different story, since those "barbarians" have much better stuff to offer than PNW and California natives. Probably would result in Chinese communities in Mesoamerica and even the Andes.