Let's break things down.
North Korea has a highly trained army on paper. I'm not sure it's the same on the ground because of the economic problems. These supply difficulties might make them devote lots of time to growing potatoes and be really good at that. But North Korea does have a good military reputation and is a major supplier of military training for African armies. I would count the professional core of the North Korean army as having a very combat proven education, giving confidence to their troops.
South Koreans have more and diverse food since childhood, this means they are on average more capable (mentally and physically) than their less fortunate brothers in arms. I wouldn't dare to compare their education systems, but because South Korea is good in comparison to other countries, I'd also say that because of better training at knowledge adaption the output of training, including military training, is amplified.
And of course, the South Koreans have more money to buy toys than the North that has to make do with other means. I wouldn't exactly count North Korea as China's pupil. They are influenced by Russia and China.
Russia and Mongolia are what remains of China's powerful northern neighbours, with all the horse-riding nomad threats becoming oblivion by motorized progress. Mongolia doesn't seem a threat on its own, but they are capable of serving as an excuse if Russia decides to support these "poor buggers" and reclaim some old Mongolia. Futhermore Mongolia is a nice geographic region for military operations, see Soviet final land battle against Japan to get a feeling of the geographic importance of Mongolia. What China and anyone else is unlikely to be afraid of, is the modern Mongolian army.
Korea, especially North Korea, has a much neglected narrative that sets them apart from China and would actually serve as the reason for them to go on a warpath with China. It's the cleanest race narrative
From that point of view it's only a matter of preparation before a unified Korea frees her nationals from other "dirty" peoples and China will be one of the places to start. Money for the political groundwork will likely be donated by industries with strong competition from China.
If Korea gets unified and the Korean border issue with China is not regarded as settled by the unified Koreans, then I would place my money on the Korean military to have the ability to get what they want by force and hold on to it. The war would be about a limited territorial grab of ancient lands (important psychological aspect) and a strong defense afterwards that exploits Chinese economic vulnerability to irregular naval warfare while keeping strong entrenched positions on land. The air control can be defended by a combination of light fighters and air defense, while China would need a several times more expensive force for an offensive achievement of air supremacy. You could also bet that the US wouldn't let the opportunity slip to test lots of new sensor arrays that give the Koreans the right information to hit and hurt back.
The large numbers of military trained males and females in Korea are likely not decisive in a human wave attack in a military conflict and most of them would serve away from the fighting zone. The advantage of such a large registration and training of military skills is that you can assemble a large force of high quality soldiers out of the available trained manpool and do have ready replacements that already know a bit about military matters, thus will be more open to the rapid wartime trainings. I count on a very capable Korean force of 1-1.5 million soldiers to start the conflict and sustain their numbers and quality even during a prolonged fight. On short notice Chinese commandos could face well-organized "militias" throughout Korea that outnumber and pin them down until better troops arrive to settle the issue. This large number of part-time soldiers in local defense readiness would only need infantry equipment and communications. I mention them because I believe WWII showed the way and countermeasures for commandos in conflicts between competitors of equal military skill.
So I bet on Korea in an armed conflict because the goals of a conflict would be more important to them than to China that could find a way to settle the issue without losing face. China and Korea could both strangle each other's economy (especially North Korean commandos are a feared force), but because of the importance of the national unity goal Korea would be willing to hold their breath for longer than their enemy.
China has to lose prestige from a possible conflict, but could win politically by establishing a precendence for armed national unity (Taiwan & ?) and by achieving a Korean neutrality (finlandizing it), thus on a global scale weakening the US position by moving much of shipbuilding capacity out of their alliance structure.
The likelihood of such an armed clash is small because the ends can be achieved by other means and likely will be unless someone in the region goes really crazy.