From Google: "U.S. goods and private services trade with China totaled $579 billion in 2012 (latest data available). Exports totaled $141 billion; Imports totaled $439 billion."
That's five hundred and seventy nine billion reasons why China wouldn't back N. Korea if they attacked S. Korea. The US is just one country with ties to S. Korea that China deals with.
China will do everything it can to prevent war between the two Koreas, but if it does break out and North Korea is the aggressor (which is ten times more likely to be the case than the reverse), then China will do absolutely nothing to support North Korea's war effort, but it will most likely do everything it can to prevent the collapse of North Korea's central government. China's strategic interest in North Korea has changed since the 1950s. During the Korean War, NK represented a strategic buffer between China and the United States and its allies. Now the United States no longer presents an imminent danger to China, and South Korea even less so. The strategic interest in North Korea is keeping it stable, because if North Korea were to collapse it would mean millions of refugees pouring into China.
If war broke out, I think China's priority would be to seek an immediate cease fire before things could get too far out of hand.
As for the matter of North Korea using nuclear weapons on South Korea, I have no idea what China's reaction would be, but I can't imagine they'd be happy.
Doubt it. The US could level every military base, silo site and key position in NK without nukes.
Using nukes is little more than a show of force for the US now and the US already has the "we'll drop a ****ing nuke, we don't care" card. Only way I see the US retaliating back with nukes is if nuking is the best option from stopping another imminent NK nuke from being launched.