Islam had reached the Korean peninsula centuries before the Joseon. The Goryeo were trading with Central Asian Muslims during the Mongol period (you know soju? That originated from arak, a Levantine liquor, after it was brought over by the Mongols). As for actual conversion, it was never very popular as far as I know.
Issue is that trade wasn't really looked favourably upon by the landed gentry (looked down on merchants) and China's vast markets are right next door anyways (Joseon Korea, being the most favored tributary, got the most access to it) and Japan's also there. Not really much of a pressing need to search for more markets or more resources since China has both.
As for Europe, the Joseon looked down on them when they did become aware of them. No Imjin War might mean more of a western presence but any reactionary king could just do what OTL Qing, Japan, and Joseon did and stamp down on westerners and Christianity.
Biggest butterfly? I'd say if mountainous Korea gets ahold of potatoes and other New World crops a century earlier (sweet potatoes came in 1764 from Japan). OTL Qing saw its population double or triple, IIRC, in the centuries after that was introduced and potatoes came from the Inca highlands, the terrain of which being more mountainous than Korea. There's plenty of space on mountains that, without terracing, can't really support rice production that could be used for potato growth. More food, fewer famines, the Joseon population grows much faster and more of the peninsula gets populated, especially the less populous north. Which, as it happens, was the region most receptive to Christianity when it did come to Korea OTL.
Other than that? Butterflies and kings determine that bit, I think.