This is a very naive analysis on your part.
1) Minimum loss of life for whom? The U.S. ?
USA
would have absolute air supremacy
after a few days. The problem is, what North Korea can do after 3-4 days? The answer is,
a LOT of damage.
http://www.csmonitor.com/World/Latest-News-Wires/2012/0104/North-Korea-military-has-an-edge-over-South-but-wouldn-t-win-a-war-study-finds
2) North Korea har between 180,000 and 200,000 Special Forces. They are Kim's private elite guard. In addition to another 1 million soldiers, North Korea has between 6 to 7 million strong paramilitary force.
So even though the U.S. and South Korea can repel any North Korean attempt taking Seoul, that doesn't mean that North Korea can't do a lot of damage to civilian infrastructure. Remember, North Korean forces are 50 kilometers (31 miles) north of Seoul.
South Korean forces are about 140 kilometers from Pyongyang.
3) Most citizens in Pyongyang are the most regime-loyal ones. In order to get permission to live in Pyongyang, you have to be one of the most loyal ones. As North Korean citizen, you can't just move to Pyongyang. You would need permission to do so. In order to get permission, you need to prove yourself over a period of several years.
Furthermore, there is nothing that indicates that any North Korean citizen would "roll out the red carpet" for any U.S. or South Korean forces. This never hapened in Somalia, it never happened in Afghanistan, it never happened in Iraq and it's not happening in Libya either. On the contrary, extremist groups are gaining ground. That's no wonder since U.S. is bombing their country to hell.
Based on the U.S./Western interventions since early 1990s, there is not one single country that is "working" properly as a country.
The Balkans, Haiti, Afghanistan, Somalia, Iraq, Libya, and now Mali - none of them are working properly. And there is no indication that Syria or North Korea will work "properly" after the West kills Assad or Kim.
4) China does not want U.S./South Korea troops near it's borders. That itself can definitely provoke an attack.
China might not defend North Korea, but China might easely be forced to create a buffer zone a few miles across the North Korean border, in order to still keep some kind of buffer against U.S. and a "new Korea". There is nothing U.S. can do to stop China creating such a smaller buffer zone, close to Chinese border.
I wish people stop using the word "trade". Countries are not more peaceful, just because they are trade partners. It's a matter of national security and "trade" will always LOSE against that, and trade will lose against geo-strategic interests.