The US fought for South Korea. they were just to help South Korea because North Korea is invading them.
They were sent into South Korea at the request of the United Nations to protect Sout Korea and to eject the invading North Koreans.
In the beginning it was NORTH Korea invading SOUTH Korea. Within months of that invasion, the US & UN are involved; in the defense of SOUTH Korea. Communist China enters the war on the side of NORTH Korea, after the US/UN advances too close to their border. NORTH Korea and the Red Chinese are communist nations. SOUTH Korea and the UN are free world nations. The USSR supplies the hardware for the communists to fight with (Tanks, Jets, Artillery, etc.).
The US sent troops into Korea from Japanese military bases where US troops were stationed to help South Korea resist the North Korean invasion . The intervention of Chinese troops widened the war .
The Korean War directly resulted in the US NOT INVADING North Vietnam.
No. The US leads the world in enforcing sanctions against North Korea.
North Korea attacked South Korea.
North Korea cannot possibly defeat the US in case of war. They can, however, do great harm to South Korea. I expect that if North Korea attacks the US directly rather than just attacking US troops which are already in South Korea, there would be an overwhelming retaliation. Remember that during the Cold War era, the US was prepared to go to war with the entire Soviet bloc, which was roughly half the world. The US has vastly more military power than it would need to deal with North Korea.
No North Korea is a humanitarian nightmare.
Yes North Korea does not like the US mainly because of the country's involvement in the Korean War between North and South Korea.
Yes the US is known as the American Empire to North Korea.
The US is not permitted to patrol North Korea, nor do they. The US military does however have military bases in south Korea as part of a treaty to protect them against a possible attack by North Korea.