Yes, the bigger problem will be when the North can put a nuclear warhead on a missile. That said, the moment the missile entered Japanese air space, regardless of what it was, they had every right to shoot it down. Also, we had every right to shoot it down if we classified it as a threat to our own security. Legally it would be a tougher sell, but doable.
He is a loose cannon, but I would argue most of what he does is an attempt to ensure regime survival, which is really not that insane. I have strong doubts he would start a war over this, and I have even bigger doubts that China would tolerate it, which has to be in their thinking.
No, that would be an act of war and play into their hands. If you establish the reasoning however where you are on sound legal footing, it will be pretty tough to lose that one in the world opinion battle.
Further, shooting it down could have shown that our missile defense system actually works, which would be a major boost to US credibility, and could have reassured Japan, and perhaps even countered their own proliferation goals.
No one in the North Korean military is going to overthrown the leader. That is simply not a real option. His son (that is thought to be the next in line) is hardly any better. If you want to wait, which is fine, the best option is to start reassuring regional allies, one major way to do so could have been to shoot the thing down. After that, if you want to take it to the UN, fine, but demand a resolution that classifies NK as a "threat to International peace and security" to show we actually mean business. Otherwise, all we will get is more of the same.