Zheng He did some incredible feats of "showing the flag" for China, but he never pulled off quite what Cortes or Pizarro did. He never decapitated a major empire in one stroke like those two men did. China never had lightning strike twice like happened with the Aztecs and Inca which solidified Spain in the New World (if their ongoing violent dominion on Hispaniola against the Taino wasn't enough).
We know the Chinese strategy was to have merchants and such settle in trading centers. At times, they could have major pull like in Lanfang (Borneo). But China absolutely won't go for the decapitation strike like Cortes or Pizarro did. It's far too risky. If they didn't too much mind that Japan was calling itself an empire (and very rarely its ruler paying tribute as the King of Japan), then they probably wouldn't care too much about the Aztecs or Inca not paying proper tribute to the Son of Heaven. Based on that alone, the answer is hell no, China can never conquer the New World as fast as Spain.
This forum has never quite realised that the Pacific is really, really big, and that there's nothing much of interest on the other side if you don't know that California is.
Nope, there's endless barbarians to fight for their furs and reindeer herds which fade into more endless barbarians to fight for their furs (although they don't have as big of reindeer herds). Unfortunately, furs are stereotypical barbarian clothing in East Asian culture, although antler velvet for traditional medicine might be a decent compensation. The biggest problem is that if China wanted to subdue endless hordes of northern barbarians (which the American Indians they meet will be classed under), they already have Siberia, and they don't need a navy to do so. That's why I think in the "East Asians on the West Coast" question, the Japanese are just as likely (even though they're mentioned far less). They just need to solve internal issues, subdue Hokkaido, Karafuto, Chishima, and go up to Kamchatka and go to North America from there. Fishing, isolation, antler velvet, and even furs might be a reason.