Horrible suggestion. You're basically handing North Korea the power to end the world. North Korea isn't a Chinese puppet or even much of a proxy, they're a lesser evil that's constantly a thorn in China's side almost as much as they're one in the side of the US. China can influence North Korea, true, but only to a degree. They've already expressed their opposition to North Korea's nuclear tests without any real impact.
The reason they haven't "dealt" with the problem is the same reason as for why the us hasn't either - as long as the current regime stands, it can't really be done without massive loss of life and without a major humanitarian disaster on China's doorstep.
Exactly. In spite of their traditional military alliance, N. Korea just represents a huge problem for China. The idea that they are just a pawn or satrap of China is totally wrong.