Much better than OTL North Korea, definitely pretty bad compared to OTL South Korea. Kim Il-sung is likely to resign at some point, or otherwise will have limits on his power which OTL he was able to remove. Kim Jong-il will remain a political figure of some note, but will probably stick to propaganda films and other artistic works. The regime might get overthrown in the early 90s, or otherwise reform like China did. Even if the ROK and the US keep Jeju and other offshore islands, there isn't nearly the justification DPRK had OTL to militarise so heavily.
Without having specialized knowledge on the issue, it's hard to see how, as the island is so small. I think if US would abandon Korea, it would abandon it entirely as Korea was so insignificant at the time. Of course it might be possible that USN would lobby to keep it as a base when SK collapses, and SK refugees grasp the opportunity to have an exile government there.
Proportionately Jeju compared to Korea isn't that much smaller than Taiwan compared to China in terms of late 40s population, and is larger proportionately in terms of area. The ROK notoriously conducted a brutal suppression of left-wing influence on Jeju in 1948-49. Combined with American troops, the remnant ROK, and refugee populations, Jeju wouldn't be in danger of falling anytime soon.
Further, the remnant ROK wouldn't just be Jeju, since it could easily keep control of Ulleung off the east coast of Korea, and probably some of the smaller islands off the southwest coast of Korea, which would be like ROC's Fujian Province, consisting of a few small islands scattered off the coast of PRC's Fujian Province. So ROK could retain a decent amount of land, and probably have a population of perhaps 1-1.5 million people.
If the DPRK reforms, then Korean unification is pretty likely. The remnant ROK is too small to develop like Taiwan or OTL ROK, and the economy would be largely based on the American military bases and remittances from Koreans in Japan and the US. So the economic disparity is not likely to be as high as OTL between the two Koreas, although ROK might have a slightly higher GDP per capita compared to DPRK (which will have a much higher GDP per capita compared to OTL DPRK).