Is it possible that, in 1866, France's campaigns against Korea end up escalating, and result with France making Korea a protectorate or colony? If so, how would this happen? What would be the results?
I think the effects upon Japan's rise would be interesting to say the least.
Well the French will need more than one gunboat...
In 1866 the French had one frigate, Guerrière; two avisos (a kind of sloop), Kien–Chan and Déroulède; two gunboats, Lebrethon and Tardif; and two corvettes, Laplace and Primauguet.
So far this all assumes the rest of Europe just sits back and watches while the French give themselves a large, populous, and fairly wealthy colony.
For all that it was less valuable than a position in China or even Japan, it's still a major shift in the Great Game. Korea was dead in the sights of Russian expansionist ambitions in East Asia, for example, and they'll not be amused if they perceive themselves cut off.
The British in turn would be highly opposed to French acquisition of Korea if the French and Russians were friendly. Unless the two were at odds, it would leave the pair dominating northeast Asia. And combined with the French presence in Indochina, London would seriously consider the risk of East Asia falling in its entirety under Franco-Russian influence. Which would, of course, be unacceptable.
Was that true in 1866? They certainly had their eye on Korea in 1904, but that's still a long way off.
In 1866 Franco-Russian relations weren't great. France had been a bit too sympathetic to the recent Polish revolt.
The great powers were interested in whatever their opposite numbers were interested in. If one got Korea, the others would want to prevent it or get matching gains.
In terms of Japan's position, if you get a competent person in charge of the French diplomatic effort they might get Japan's support if they do something like offer them equal status (to France) in terms of trading, resources and related matters.
In general though, I don't think the other Great Powers, save Russia would care that much, none of them but Russia had any interest in North-East Asia (apart from China as a whole) by that point, so apart from ensuring the French don't dominate ALL of Eastern Asia or China, I doubt they'd do much.
Now, having said that I could see Indochina developing differently; during the time period being talked about here France only had a very small portion of the region, and it was a very recent gain (almot all of Frances gains from then on were after 1885), and with Korea they may not be as interested in expanding to the same degree, so for instance France may only expand at the expense of Annam and Luang Prabang and not take the bites out of Thailand that they did IOTL.