how likely is a US war with NK?
There's a link to a podcast about North Korea in the international politics thread that adress the issue.
To paraphrase, it is not very likely unless / until someone fucks it up.
In short :
The USA knows that NK can bring a ton of artillery to bear on Seoul in minutes.
NK knows that initiating hostilities will start a war they can't win.
The plan outlined in North Korean propaganda (the one for domestic audiences) is to ultimately call the USA bluff by escalating on their nuclear capabilities and force them to retreat from the current very extensive guarantee while simultaneously not scaring off South Koreans away from the "Sunshine policy". Then hopefully to start a political process for Korean reunification that they hope to impose on their terms.
Like all such plans, it works fine on paper by assuming everyone is rational to an extent.