Yeah that's the problem you need the Daewongun to break the Andong Kims but then once he got power he wasn't going to pull a Meiji. Maybe have the Daewongun die earlier and have Queen Min consolidate more power without the distraction of her father in-law trying to push her out? That might be workable but might be too late to do much good? Maybe if you put in just a slight Japan screw to keep them from kicking around the Chinese and Russians too badly?
If the PoD is too early, this only allows either the Andong Kim clan or another conservative faction to take over. If it is too late, then it's going to be extremely difficult for the monarchy to effectively maneuver the factions. As an example, the Gapsin Coup was initially successful because of officials actively pushing for industrialization, but the queen was forced to seek aid from China in order to push the reformers out. There might be a compromise PoD, but the queen would then be too young.
Then for simplicity's sake, maybe it would be best to simply have, say, the French be more aggressive and forceful in their incursions. To bolster French morale, let's have them have taken Indochina by the mid 60s. Like I said, Korea doesn't really have to be a good idea, the idea of colonizing it just has to have appeal. The French did take the trouble to attack it once IOTL after all, just not with enough troops. It should be feasible make a PoD that would allow for France to feel the need to use more force.
Well the French were distracted by the intervention in Mexico. Resolve or remove that, and they might be more focused on the Pacific.
The French intervened because Korea had tortured and/or executed several French missionaries, and they actually managed to smuggle out more than 250 volumes of court records (Uigwe), so the intervention was somewhat significant. In addition, the French were mostly defeated in Mexico by 1866, so if they had wanted to conduct a more thorough expedition, they probably would have done so later on. The fact that only a few were sent to Korea in 1866 suggests that the French were simply not interested in the peninsula, and that it was considered to be a relatively minor affair. Also, sending more troops to Korea might lead them to temporarily take the peninsula, but it would require a significant amount (probably around 10,000-50,000) due to terrain and stiff Korean resistance, when they could be used elsewhere.
I do believe you mean 1866 and 1871.
Fixed.
The issues might be similar, but Korea had to deal with several factions favoring China, Japan, or Russia, not to mention that the government generally remained strictly isolationist, which is unlikely to change. In fact, the Opium Wars only convinced Joseon that it should tighten restrictions so that it would not be "run over" by foreigners.
They would be interested. Russia however has a big advantage. It has the actual land border and its huge influence in Manchuria.
Russia tended not to play by quite the same rules as other nations (it did to an extent of course, but it was always quite the bad boy). It would be all too easy for Russia to launch a legally dubious invasion.
It took the Russians about two centuries to consolidate its holdings near the Pacific, and it wasn't until 1858 that they were able to expand into Outer Manchuria, which was only possible after the Qing had suffered crushing defeats during the Opium Wars. Russia was also concerned with other internal and external affairs, such as political reforms and factions, along with military conflicts in Europe, so I don't think that attempting to actively expand into Manchuria and/or Korea would really be its main issue at the time.
How about No banishment but given a settlement and allowed to teach starting a school
I'm not sure how this would occur, not to mention how reforms could be widely implemented.
Maybe if this group had brought more than just western religious philosophy.
I think you're aiming for records concerning agriculture, but the problem is that bringing too much over would attract suspicion from the Korean government. Korea had implemented a significant amount of reforms concerning agriculture by 1700, but without infrastructure to complement these changes, the effects would be limited at best.
I would like to see Queen Chongsun survive longer and listen to Chae Che-gong removing the book ban for western books imported through china.
Not entirely impossible, but the majority of the factions would be fiercely against foreign influence, so I find it hard to see how this could be butterflied away.
Best one here would be no restriction on travel aboard and this leads to korean connections to the kingdom of hawaii
Another possibility is the crew of the General sherman are forced to abandon the ship as it sinks in the river before the incident leading them to be forced to ask for korean aid
As stated before, there is no particular reason for the colonial powers to actively intervene in Korea, nor the Korean government to open up under Heungseon Daewongun.
Same shipwrecks force the Americans to ask for korean help. Also following the koreans manage to recover the sunken ships and try to study their design.
Plausible, but studying a few ships aren't going to lead to widespread changes, not to mention the near impossibility of dredging them without significant damage. The Koreans would probably regard them as curiosities, but their interest would most likely be limited as a whole.