1) We can say the Cold War began in 1947, with the apparition of Truman’s containment policy. In 1950 it was the very end of the Berlin blockade, in which the Russians and the Americans challenged each other by means of Berlin’s people, and which lasted from 1948 to 1949. 1949 is then marked by the division of Berlin in two parts: the RFA at the West, and the RDA at the East. This last one is led by the Soviet Union, whereas the first one is led by the free Western Europe plus America and England. This achieved to divide the world in two sides, even if from 1946 Churchill considered that Europe was divided by “an iron curtain”. This same year, in big part thanks to Mao, China becomes communist. Then, when the North Korea (Communist) invaded the South Korea (under US’s protection), America had to react, and, in 1950, broke out the Korean War.
We can see that the tension accumulated since several years between the two sides just exploded in 1950, and that there was therefore a very high tension dividing more and more a world already bipolar.
2) The US cared about Korea because of their containment policy (see question 1). Indeed, they were afraid that Communism would be as when you push a domino and that one fell, then another, then another… They though (or at least the people did, while the politicians seemed and only seemed to think) that if they let Korea be communist, it would infect the all free Western Europe. Moreover, as the South was under America’s protection and the North communist and then related to the Soviet, it was also a way for each of them to defend not only a part of a country that wasn’t theirs, but also their ideas and ideology.
The UN did play a role in this war, and yet it wasn’t interested in Korea at all. But during the meeting to decide if forced had to be send or not, USSR was absent. Indeed, they boycotted the UN because China was fired of it (they considered than Mao wasn’t the legitimate governor). USSR was absent, then, and it