I was reading up on the Soviet Japanese war, and wiki told me that Russia had invaded Korea so quickly, that they could have easily established a presence across the entire peninsula in a short period of time. The only reason they didn't (from Wikipedia) is that America just asked them to stop at the 38th parallel. So they stopped and let America take over the rest about a month later.
Why would they stop when they could have had the whole thing plus China?
South Korea wasn't established at the end of World War 2 and the United States didn't establish South Korea.
South Korea became a country on August 15th, 1948 (Happy Birthday ROK!) after the election of a government earlier that year. From 1945 until 1948, the southern portion of Korea was under US supervision and the northern portion of Korea was under Soviet supervision. The United Nations proposed nationwide elections for 1948, the Soviets (and North Koreans, who were quite strong-willed without the Soviets) realized that they wouldn't win (the southern population was much larger than the north) and boycotted the election. They instead formed the Democratic People's Republic of Korea (DPRK), or North Korea while the south held elections and formed the Republic of Korea (ROK), or South Korea.
Why did the Soviets allow the US to take the south?
Also, you mentioned,
Why would they stop when they could have had the whole thing plus China?
The Soviets DIDN'T have China. The Chinese had been fighting a civil war from the late 20's until 1950, the Nationalists and the Communists. They united to fight the Japanese, but once they were defeated, almost immediately, the Civil War began again. The United States supported the Nationalists with money and weapons. Until the collapse of the nationalist government in 1949, US influence in China was much larger than Soviet influence. Even after the success of the Communists, Stalin and Mao did not have the best of relations. Throughout the 50's, the Sino-Soviet split grew until it happened in 1960.
Long story short, it's easy to look at it from this point and say "what the heck were the Soviets thinking," but at the time the cold war had not started, America and Soviet interests were both focused on Europe and the Japanese home islands, and China was busy with their own issues.