North Korea can wipe Seoul from the face of the earth with conventional weapons quickly if they stuck first. We would most certainly get involved and it would be a long and drawn out war that would approach Vietnam level casualties if we invaded.
Again, the key is whether they have Chinese support. PRK by itself cannot sustain major combat operations for very long at all and would be quickly crushed even assuming the soldiers are dedicated to the cause, which becomes less likely over time. This would not be close, there might be minor skirmishes north of Seoul with some civilian casualties, but nothing of strategic importance.
If China were to actively back the PRK like they did in 1950 it could be another animal, but we are on more friendly terms with the Chinese these days so the fate of North Korea is more likely to be settled over a table than over a field.