Let's start with the last question. Where there any Asian powers capable of threatening the USA in its infancy? The answer to that is simple: No.

This is easily answered because the USA in its infancy was made up of 13 colonies and unorganized territories on the East Coast of North America. An Asian power would need to not only get to the Americas, but they would have to traverse the Indian Ocean, round the Cape of Good Hope, and then sail north through the Atlantic to even reach the infant USA.

And a single trip would not be sufficient, an Asian power would have needed some sort of ability to supply and maintain an colonization force from their home territories through territory that was already under control of European colonial powers.

In short, the logistics would have been insurmountably difficult, and there would be powerful European enemies to overcome even before an Asian power could even commit those sort of resources to a late 18th or early 19th Century attack on the USA. While it would not have been completely impossible, the committment to this sort of endeavor would have had to have been total, very long term, and would have effectively started the first World War to even get close.

I suppose it is possible for them to have landed in what would be California and colonized and then eventually moved across the continent from the West, but they would have still had the logistics problem and would have still been directly opposed by the Spanish, and possibly the French in Louisiana in their ambitions to deal with the young USA.

Moving on to the question of Asian powers settling in North America, we have to define who was capable of this at any point before the 19th Century. The answers to this would have been either India or China, but if you consider Russia to have been an Asian country, it would have counted as well.

To start off, I should point out that for the most part, the Asian countries capable of exploration completely lacked a motive to actually do so. The reason for this is that they often viewed themselves as the richest and most desirable place on Earth already. This high opinion of themselves was not without very good reason.

From the Fall of Rome, to the middle of the 19th Century, the largest city in the world tended to be a Chinese city with only a few interruptions.

China and India were already fabulously rich and had high populations throughout most of the period between Ancient Rome and the settlement of the Americas. Most of the trade in spice, tea, silk, porcelain, and other sought after commodities originated in Asia.

In fact, until the Industrial Revolution, the Western world maintained a significant trade deficit with Asia as they had no products that were considered desirable to Asian markets. This required the West to simply send massive amounts of silver to Asia to pay for these products directly.

It could be argued the most important reason for this failure to reach North America was quite simply due to lack of motivation to do so, but there were other critical reasons that this might not have occured as well.

The first issue was the difficulty of reaching North America from Asia. The Europeans had a relatively short trip accross the Atlantic Ocean to the Americas, and even so, this trip could take weeks or months.

Reading the Americas from Asia had two possible routes in general, both relatively disadvantageous.

First, they had the option of simply sailing west like the Europeans did. This would be relatively simple to do until they reached the coast of Africa. Upon reaching Africa, they then had to round the Cape, and sail north. At that point they might be able to sail West to hit South America, but chances are good, they'd hit Europe before they managed to actually take a direct route to North America. So, this is a really, really, long way to go. And chances are really good that they would have found quite a few more interesting things long before they even tried to cross the Atlantic Ocean. Such as a highly developed European continent.

The second option was more straightforward but similarly difficult, as it required them to navigate the Pacific Ocean. The Pacific Ocean is much, much wider than the Atlantic. It would also have the disadvantage of being almost completely devoid of places to land for supplies once they were in the middle of it. If Columbus had needed to sail the Pacific instead of the Atlantic for his voyage, his men would have mutinied and turned around long before they even got halfway. European explorers like Magellan certainly managed it, but the trip could take years and these expeditions often had signficant losses in men and ships.

All of that being said, it is clear to me that an Asian power like China did have the sheer technological and economic might to at least make a discovery of North America, if they had the motivation and an inkling that something was waiting for them across the sea. But they needed the critical motivation and the right experience to make such a voyage possible. And they needed a little luck on top of that. East Asia is poorly positioned for a discovery of North America, Europe is relatively close in comparison, but there are exceptions to that.

Remember when I mentioned Russia? Well if you consider Russia to be an Asian country, they did discover parts of North America. Alaska to be exact. And they settled it in 1741, before the foundation of the USA. This probably doesn't really count for your question because (a) you may consider Russia more of a European power and (b) Russia already knew North America was there when they started looking.