a. We won Korea, because we saved South Korea from being overrun, which would have happened had we not intervened.
Just to be thorough I am going to look at each major player and assign them a Win/Loss/Neutral.
South Korea: Win. They retook their country and set themselves up for heavy U.S. aid that made their country prosper.
North Korea: Loss. They lost tons of men, had their countryside ravaged, all in order to become a Chinese vassal instead of a Soviet vassal.
China: Win. They took massive losses, but repulsed the Americans and established themselves as a regional power.
USA: Win. We took relatively light casualties, proved our willingness to assist allies, which strengthened our position internationally, and field tested a range of military equipment.
UN: Win. Proved its effectiveness in taking action against the North Koreans.
b. North Korea is no threat to either us or South Korea militarily. However, their nuclear capability and unstable leadership is.