This is irrelevant. To begin with, the Qing invaded Joseon in 1627 and 1637-8, both of which only lasted for 2-3 months, and the campaigns were conducted specifically in order to prevent Joseon from aiding the Ming. While Korea was undergoing severe political turmoil at the time due to the Imjin War (destabilizing the military), the opposite was true for both cases by 1871, as Korea had experienced well over 200 years of relative peace (with the exception of a minor French campaign in 1866), causing the military to deteriorate over time. However, the peninsula was able to remain independent because the European powers were disinterested in Joseon, which I will go into more detail below.
See above. Additionally, the Manchus (Jurchen before 1635) were able to force Joseon to pay tribute because of the devastating aftermath of the Imjin War, which forced the country to invest resources into fortifications in the south, as well as extensive rebuilding efforts after the invasion, temporarily diluting the main focus on the north, which had been continuously maintained for over a millennia with troops of several hundred thousand across various fortifications. The situation was further exacerbated after a palace coup in 1623, in addition to an uprising ten months later by a general who had made the coup possible in the first place, causing the monarch to temporarily abandon the capital. While the rebellion was put down within a month, some rebels fled to Manchuria, then persuaded and aided the Jurchen in invading Korea in 1627. While Joseon attempted to make preparations for another potential invasion, the military and fortifications generally continued to remain in disarray, allowing the Qing to quickly seize the initiative in 1637.
Joseon had also been making preparations for a potential confrontation soon after Gwanghaegun came to power in 1608, in addition to utilizing diplomatic policies to stall a conflict, along with consolidating the military both before and after the second invasion. As a result, Korea would have been much more prepared for an invasion had the Imjin War not occurred, Gwanghaegun not been overthrown, or the Jurchen/Manchus had invaded over a decade later, which could have then allowed the Ming to regain the initiative.
Also, for reference, the Manchu ruler at the time should be referred to as Hong Taiji (Huang Taiji), instead of "Taizong of Qing." This is because Shunzhi, his son, was the first in his dynasty to consolidate rule over China, while Qing rulers starting with Shunzhi (except for Puyi) are traditionally referred to by their era names, instead of their temple ones.
Yes, but the European powers were collectively disinterested in Korea for over three centuries, despite a continuous interest in China and Japan during the same time period (although with some limitations), which was mostly due to the fact that Korea was not located near any major trading routes. I've also discussed this extensively elsewhere.
However, had the US indicated an active interest in Korea in an alternate scenario, it would have been far more likely for other European countries to do so long beforehand, given that they had been in a much better position to do so for centuries. Such divergences would require a PoD to occur at least by the 17th century or so in order for the peninsula to become more attractive to European trade, which would also predate the foundation of the United States. In fact, any significant foreign military involvement by the 1860s-70s (as a result of centuries of divergences) might actually incentivize Korea to pursue industrialization, or at the very least militarization, essentially contradicting the OP.
It's also telling that IOTL, Japan never directly fought the Korean army, but rather influenced the country gradually over several decades before it was eventually annexed.
