In the past Hitler started WWII over gaining "lebensraum" or "living room" for the Germans. I see the same here, although the environment is different, and not very conducive to a tragedy on the scale of a war.
Actually, Hitler started the war because of the Polish planned invasion to Germany and the secret superweapons that the Polish were developing. Which was pretty much the same "reason" as U.S. imperialist aggressor troops threatening North Korea.
#1: Does he really believe he could win a war? Do his people really believe that?
Yes, no, maybe. What does it matter to a nutter? Besides, death in a war destroying the U.S. imperial aggressors is a death with honor... so... "winning" is not really all that important. Even if you "lose", you still win because you destroy the evil guys.
#2: Where do you think he will strike first?
A "first" strike would have to happen in several locations to be effective at all.
Japan and South Korea would be logical targets for nuclear missles. U.S. mainland, presumably with biological/chemical attack and saboteurs (destroying infrastructure, poisoning water) would be a logical move. A simple blackout that covers a dozen major cities probably causes more destruction than a nuclear missle, and it's practically impossible to counter.
A handful of "anti-imperialist warriors" with bazookas can probably take out a major city in minutes, no problem. All it really takes is destroying the 2-3 biggest relay stations in the region, overload will take care of the rest. For bonus points, fell a few pylons if you like, or break a dam. That's blackout for at least several days (or weeks), with people panicking, looting and killing each other in the street. The military will be busy shooting at their own civilians to keep up civil order, they won't have time to shoot at you.
Add to that whatever a few dozen skilled saboteurs/terrorists can do in their free time (put some explosive on the central pillar at the local hospital, cut phone cables, bring an airplane down into a building... you get it). There's absolutely nothing you can do against it, except shoot anyone who looks remotely Asian on sight -- and even then you can't be sure.
(On that note, consider how you're being scanned, screened, biometrized, stripped naked, and being nazi-treated in every possible way at every airport. At the same time, almost all the guys driving fuel trucks on the airfield and almost all guys loading/unloading suitcases have a rather Mullah-like appearance and in fact look just like they could be Bin Laden's younger brothers. Are you feeling much safer now? You're welcome. Now consider how many Koreans work in that nearby nuclear power plant and in your local hospital, and in the fire department. Of course you can be 100% positive that they are all "good Koreans".)
Attacking Hawaii, on the other hand, would be a really stupid move, but mentioning it may lead to divert the enemy's forces and sustains fear, uncertainity, doubt.
There is nothing on Hawaii worth destroying, the U.S. have a lot more missles in submarines (or in other countries) than on Hawaii, the same goes for troops. And the few civilians living there are hardly worth the rockets that will take a long time to reach their target and will likely not hit on spot anyway at that distance.
Japan and SK are much better targets. Short range, missles likely to hit, high population density, not much of an advance warning. That is, low chance of antimeasures, maximum casualties. Plus, destroying a rich industry.
#3: What will he strike? Military bases only?
Striking military bases is nonsense. The U.S. did not win World War II by striking Japanese military. They won by killing civilians. Never in history have wars been won by killing soldiers. You win by killing civilians, raping women, destroying houses, burning crops, and salting the earth.
Kill morale and supplies first, then worry about soldiers.
#4: How will he strike?
First, he will strike by talk and threats for years to go, until nobody takes him serious any more.
Then, assuming he hasn't been murdered by some even more insane competitor (or by the CIA) in the mean time, my guess is that he'll come up with a coordinated sabotage/terror strike on U.S. mainland, and nukes onto Japan and South Korea at the same time.
#5 and #6:
Impossible to say, even speculatively. The U.S. might just wipe out NK within 6 hours, or they might not as to not risk hitting nearby allied targets or provoke a counter-strike from China, who might just see missles launching and hit the red button, as a reaction.
It's not like you could fire missles at NK without directly and indirectly hitting two nearby allied countries, and without pointing a missle head into China's direction (which they surely won't appreciate).
Doing a "surgical" counterstrike is wishful thinking, there's no such thing. And doing a nuclear strike at all is something you don't do lightly if you're still a bit at your senses, regardless whether you've been attacked or not.