There are parts of your rather thoughtful assessment that I agree, and parts that I disagree with.
they fear even more a united democratic Korea that might (who knows?) have US troops stationed in it near the Chinese border.
I'm not sure that's the fear. A unified Korea, assuming from the South, would no longer require U.S. troop presence. And if the South managed to unify Korea, they'd be more likely to kick the U.S. troops out than to keep them there. The only reason the South Koreans tolerate the U.S. is because that keeps North Korea out. Keep in mind that South Korea (and Japan) are not grateful for the U.S. presence. They tolerate it and only because they have to. Taiwan is the only one who's amicable to the relationship, and they're growing closer and closer to the mainland every day (they'll still like Americans, but they know the money's in China). But there's no U.S. base in Taiwan either.
They don't trust North Korea to maybe not use a nuke against them in anger or by mistake as their missile systems might simply go the wrong way and blow up in China by accident.
You have to understand that China's need is economic growth at the moment. China is afraid of North Korea provoking war against the South. They're not so afraid of a unified Korea under the North regime if it just ended there. However, if the North ever took over the South, the next logical step would be to attack Japan. And this is especially true if North Korea had nukes. There would be absolutely no restraint from the rabid war dogs in the North against Japan. You do not understand hatred until you speak to a Korean about the Japanese (even the South Koreans, who are friendlier than their batshit crazy cousins up north).
Such an action (the war, obviously) would destabilize the region enormously. The U.S. would be involved. China would be involved. Russia would be involved. Even India and much of Western Europe would be dragged into the conflict. That's the last thing China wants to see, because there's a lot of risk there with little to no reward. The risk is greater U.S. or Soviet--I mean Russian--influence in the area after the war concludes, or even of MAD.
Even if North Korea magically discovers the ICBM and hits the U.S. with nukes, China would have lost, because the U.S. is really fueling the majority of China's economic growth. Now, when China has entered a period of economic self-sufficiency, their tune with regards to a nuclear North Korea may change. But for now, North Korea is a massive sore point for China.
the Kims aren't getting rid of their nukes because they believe that their family survival depends on it.
After Bush put Iran, North Korea, and Iraq in the so-called "Axis of Evil", are you surprised? Iran is also seeking nukes. And don't forget Pakistan, which is probably more unstable and more hostile to the U.S., was not included in this list for one very big, radioactive reason. The survival of the North Korean (as well as Iranian) state does depend on it. Hell, if Libya or Egypt or Syria had nukes, the western powers would have been helping the government, not helping rebels fight against it.
The only ways that North Korea is ever going to be nuclear free is that either the US is going to attack them and gamble that they can destroy their few nuclear missiles before they leave North Korean airspace or (much less likely) the regime will collapse quickly for some unforeseen reason and the new government will get rid of the nukes.
I'm not sure you get how other countries feel about the U.S., in particular those under the "Axis of Evil" label. The only way North Korea will not seek the bomb is if the U.S. implodes and collapses on itself. Full stop. Same with Iran.