I cannot speak to the Norse settlements in North America myself, for which I would direct you to answers by /u/sagathain such as this, this, and this. What I can speak to are the claims that Zheng He reached the Americas in the 1420s.

These claims are not 'seemingly debatable', they are bunk. Anyone who is asserting the truth of these claims is perpetuating a hoax that has been kept afloat by conspiracy theorists of various stripes since the publication of Gavin Menzies' 1421: The Year China Discovered America in 2003. Menzies had no formal qualifications in history nor a known grasp of any Chinese languages; instead he was an officer in the Royal Navy – one who retired under rather ignominious circumstances in 1970 after ramming a US Navy minesweeper in the Philippines in 1969.

Menzies' work is shoddy in the extreme, with its only evidence for a pre-Columbian Chinese arrival in the Americas being based on maps whose dates he can only offer poorly-grounded assertions for; superficial analysis of genetic evidence; and supposed archaeological findings that he often provides no citation for. Menzies' book was so comprehensively bad and wrong that an entire website was created by a group of academics responding to it, 1421exposed.com. Today the domain has been bought up so don't copy the link straight into your search bar, but most of its contents remain archived via the Wayback Machine. Menzies would later claim that a Chinese fleet sailed to Italy in 1434, in a book unsurprisingly titled 1434: The Year a Magnificent Chinese Fleet Sailed to Italy and Ignited the Renaissance, an even more ludicrous book which was also responded to on 1421exposed.

The specific Chinese map that supposedly proves Zheng He's voyage was 'discovered' by a Chinese collector named Liu Gang in 2005 and publicised in 2006, and was purportedly a 1768 copy of a 1418 original. This too has come under considerable scrutiny and is almost certainly a fake that was produced specifically to support Menzies' claims post-publication; while it was supposedly carbon-dated, the sample of paper supplied to the laboratory cannot be confirmed to have actually come from the map; moreover, even if the sample was authentic, and the map dates to the mid-18th century, it fails to prove the existence of an early 15th century original.

Moreover, Menzies doesn't even stop at the claim that Zheng He simply reached the Americas. His book argues, among other things, that the Chinese fleets charted the west coast of Africa, both the Pacific and Atlantic coasts of the Americas, and Australia and New Zealand for good measure. Even more offensively, he (spoilers because TW: sexual assault) >!<. I cannot begin to stress how horrific this book is; so much as entertaining its claims means giving legitimacy to a whole host of nonsense that is overtly and deliberately offensive in many parts.

The main things to look at are as follows:

• General debunk of claims in 1421 by the late Victor Prescott, professor emeritus of geography at the University of Melbourne: Focus is placed on Menzies' claim that the 'Mahogany Ship' wreck spotted in early colonial Victoria was Chinese; that European maps of the very early 1500s accurately depict the Pacific coast of the Americas; claims about the supposed route of Zheng He in the Caribbean that contradict known facts about sea levels and winds in the region; and his tendentious claims about the identifiability of now-lost shipwrecks.

• Bullet list of issues with Liu Gang's alleged 1418/1768 map by Geoff Wade, independent scholar. There is a more detailed article also by Wade.

• Detailed statement on issues with the 1418/1768 map by Gong Yingyan of Zhenjiang University, arguing that even if authentic to 1768, the map must have been based on European examples.

There's pretty good radiocarbon dating on Venetian glass beads found at Inuit camps in Alaska. They can be dated, with high confidence, to the middle 1400s -- a couple of generations before Columbus. This may sound crazy, but actually makes a lot of sense, and here's why.

For many centuries before Columbus, Europe traded with Asia at a deficit. Europeans wanted silks and spices, but they didn't have a lot of goods that Asians wanted in. So they had to buy those silks and spices with money -- gold and silver coins. As a result, over centuries, gold and silver drained eastwards out of Europe. By the 1400s, this had become a real and pressing problem; google "bullion famine" or "great bullion famine" for details. The shortage of bullion was one of the motivations pushing the Portuguese south along the African coast -- they were looking for the source of the West African gold that came north across the Sahara.

The Venetians came up with a different trick: they started producing beautiful, high-quality glass beads. This was one of the few areas where Europeans (or at least Venetians) enjoyed a modest technological superiority; Venetian glass was the best in the world. So the Venetians leaned hard into glass beads as a trade good.

And this worked great for a while! The Venetians could produce lots of different kinds of glass beads -- different colors, shapes, you name it. They weren't gold or even silver, but they were better than nothing. Unfortunately, the market for glass beads wasn't infinite. After a few decades, everyone in Asia had all the glass beads they would ever want, and their value crashed. This was bad for Venice, but it did serve as a great help to future archeologists. Because there was a one-time pulse of glass beads, outward from Venice to the Middle East, South Asia, and eventually all the way to China and Japan. These things last pretty much forever, and they turn up in graves and hoards and archeological sites of all kinds, and they're an excellent resource both for dating purposes and for piecing together trade links.

So it was surprising, but not really shocking, when a few of them turned up in a pre-Columbian site in Alaska. We know that lots of Venetian glass beads got as far as China and Japan. And we know that the Alaska Inuit traded with the Aleuts and across the Bering Strait. So, presumably some beads went something like China-Manchuria-Siberia-Aleuts-Inuit. That's a long way, but all the individual trade links in that chain are well attested. Very probably other stuff was being traded besides those glass beads -- but the beads have the advantage of being very portable, and also pretty much indestructible as long as they're not broken.

So, yeah -- decades before Columbus, there was definitely indirect contact with East Asia up in Alaska.