They are posturing. They are living in another era, on another planet. Kim's grandfather, the first 'president', is currently considered the president for ever, even though he has been dead for centuries. They need to grow up, get a grip, and enter the word. At one point or another, they will.
South Korea is not threatening anyone with nuclear weapons; it is North Korea that is doing the threatening. The motive behind the North's activities is to press for humanitarian aid using the threat of nuclear war as incentive for the West to provide this aid (like a ransom).
Some consider North Korea to be part of a group called the "Axis of Evil" who are out to do damge to the United States. They believe that if North Korea starts producing a-bombs themy might sell or give one to terrorists who would try to sneak it into the United States and detonate it.
Most probably, the threats are an excuse to keep North Korea's people united against aforeign enemy so that they won't rebel against their government.
The US isn't really doing anything in North Korea at this time. There are no formal diplomatic relations between the two countries; North Korea is still technically at war with the US and South Korea (although the US never actually declared war on North Korea). North Korea is notorious for its desire to keep foreigners out, and its borders are well guarded. North Korea also refuses to accept trade or even foreign aid from the US. However, North Korea has a large military and nuclear weapons, and their friend and neighbor China is very powerful, so the US does not have the option of militarily attacking North Korea. So there really isn't much going on between the two countries, except occasional threats or small scale military actions. Hatred of the US and its allies is a major propaganda that the North Korean military uses to keep itself in power, which is why you often hear about North Korea threatening the US, South Korea and Japan.
No. The US leads the world in enforcing sanctions against North Korea.
North Korea attacked South 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.
The US was part of a United Nations force fighting North Korea and its allies.
Primarily for provocative statements he made, threatening the use of nuclear weapons against the Chinese and North Korean forces.