1)
I can't imagine how we could engage the world in a way which increases justice without also increasing violence. We have to ask ourselves if it is worth conflict to not support a regime. For example, if we choose to oppose regime X because of its treatment of its citizens we can either isolate it or push for regime change, but these both have costs - and often high ones. If we take to pushing for regime change for nation X then what of nations Y and Z? Do they accommodate us or further repress their populations IOT remain in power? WE can isolate nations who fail to provide adequate justice, but that depends on the willingness of all global powers to behave similarly. A repressive regime doesn't necessarily need us but, but we might need them. For example, oil producing countries must remain as stable as possible IOT keep the cost of oil down. Even if we don't get our oil from country X, instability there still effects global prices which in turn enrich those who oppose us (Russia, Venezuela, etc.) and potentially reduce global availability - which in turn can lead to conflict as global powers race to secure what they need.

So I guess the short answer is we promote justice always through soft power and setting an example (as best we can anyway), but before using "tougher" methods, we evaluate the overall cost...

2)
I don't think we have an order to restore - the system is more like a poorly maintained machine which requires significant repairs but which we cannot live without long enough to do more than quick field repairs.

I don't think America can do much outside of direct, costly, and painful interventions short of trying to play by the rules and working to maintain fairness in the way resources are extracted and then distributed or things are manufactured and then sold.

3)
Yes. The United States wants to be seen as Captain America but I think in many cases is perceived more as a Jekyll and Hyde. At best, we are probably perceived as being mildly schizophrenic and at worst, severely so with a dose of megalomania added in for fun. I'm not certain those perceptions are valid - but I can understand where they come from.

4)
To be the big stick. Sticks don't talk - they just sit in a corner and look imposing.

That's an oversimplification of my view I suppose - I support the use of the military for humanitarian ends when necessary. Sometimes we can just get there more quickly than anyone else.

5)
I don't know Mike - I'd like to see more money invested in State and I'd like to see more options for officers and state to go to each other's professional schools, but that's only going to help them work better together. The real problem is, I think anyway, that all of these organizations view each other as rivals for parts of the same pie; and we contribute to waste by having a system which determines budgets based on what you spent the year before. If we resolve the turf and budget conflicts, we can probably start making real progress towards fixing the rest of the problems. Until then though, I think institutions are going to dig in and resist change for fear they'll also lose influence and money.

6)
I've thought about this alot... and I don't think I have a good answer, or at least not a defensible one so I'll leave this question alone for now.