"1. ...how do we creatively engage the larger world so as to increase justice?"
That's a dangerous and leading question. How do we American xenophobes, insulated as we are, determine or even have the right to talk about increasing justice? Moot question. Our political system will provide answers (multiple) that not every one will like...US domestic political concerns rule and the players change constantly. That is by design and it generally works acceptably if not well for us. It almost never works well for the others in the world.
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'Justice' elsewhere as envisioned by the question posed depends thus totally on US domestic politics and there will be no constantcy. All things considered, our mixed record on this score will continue and we will probably continue to do a bit more good than harm -- hopefully recalling that when we decide to do harm, we often get carried away. We just need to develop a bit of restraint...
"2. What can the United States actually do to restore order to the world without having to engage in either global policing or nation-building?"
Aside from the raw arrogance of the question, from the fact that it is NOT our job to 'restore' such order -- I question that it has ever existed... -- and from the naivete disclosed by the question, it serves as an exemplar of why the answer to question 1 and particularly to question 3 are resigned chuckles. Unfortunately, that naivete is shared by far too many in the domestic political arena and in the Foreign Policy establishment -- not to mention the Armed Forces. That naivete combined with overlarge egos and arrogance in those establishments are the factors that drive us to inconsistency in many things, to dumb interventions we did not need to undertake and fuel resistance to changes in the budgeting system which cause us to stay off balance much of the time.

Who defines "order?" What are the parameters; what happens if it is achieved to the satisfaction of 'A' then 'B' dies and the system tilts to unstable...

A better question is 'What can the United States actually do to adapt to and mesh with the world without engaging in either global policing or nation-building?'

We can best adapt to the world if we get ourselves squared away -- for a variety of reasons, we need some changes in a good many respects and areas of endeavor. We can improve our capabilities and our image if we do that. We are perceived as hypocritical because we advocate sweetness and light and yet launch assaults here and there. We can do better. we can stop trying -- and trying is appropriate -- we do not do nation building or global policing, we do selective things along those lines that suit us and those not well...

We are big, we are powerful and we are truly our own worst enemy.Those are two things we do not do at all well, do not need to do and with which we continue to play even though both are proven inimical to our interests almost always. Again, that naivete and the budget process are along with egos the culprits.
"3. Are their (sic) gaps and disconnects between what the United States says and what it does, how it wants to be perceived, and how it is perceived?"
Fuchs and Dayuhan have it right. We're a joke in the eyes of many...
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That said, as noted in the comment on question 1, above, our political system is inimical to world norms today. I would not change it so I believe we have to accept we will always be perceived as reactionary, slow off the mark, prone to indecision and, as reality does not accord with the soaring rhetoric of our Politicians who tailor their speeches to the domestic audience and forget -- or ignore -- the potential misunderstanding of foreign audiences. I might add that I've had a number of foreign acquaintances over the years comment that our domestic media and entertainment industry are part of the problem. The portrayals and the media do not sit well with many and tend to make us look like a collection of clowns. There's a lot of misperception due to all those things and there's little we can do to fix it -- short of draconian political fixes and those are not going to happen.

We'll just have to keep plugging along and try to more good than harm -- most in the world will grudgingly admit we do that...
"4. What should be the United States military role in foreign policy?"
To advise the NCA and DoS (and it incumbent upon those folks to listen to that advice. As an aside, they might be more prone to do so if we'd slow down our rotation of key personnel a bit).

Get the CoComs out of the arena. They are overly involved in foreign policy because Goldwater-Nichols gave them the ability to do that and our totally dysfunctional budgeting system so beloved of Congress to buy votes force feeds them more money than they need and it is to their advantage to enhance that flow. There's a bit of hyperbole in that but just a tiny bit.
"5. Outside of the United States mlitary (sic), what other institutions MUST be fixed in order for the United States foreign policy to be successful?
The US electorate.
6. What reforms are needed within the United States military?
That they can control: More selectivity in accessions for fewer but better quality people; Improve initial entry training, Officer and Enlisted, all services; work with Congress to improve the personnel system, reduce grade creep, refine pay, stop allowances, vest retirement earlier, encourage active and reserve crossover service, slow the rotation cycles; reduce the stifling bureaucracy; relearn how to trust and delegate -- we seem to have forgotten how to do those things...

The Armed Forces of the US basically know everything I just wrote. The question should be why aren't they doing something to correct the wrongs of which they are generally aware...

ADDED for this post on the Council:

I strongly disagree with American Pride's responses to questions 5/6. National Service is a terrible idea. Aside from the questions of involuntary servitude and 'addressing problems' that are generally transitory, the complexity and cost of administering such a program in peacetime and with lack of an existential threat is monumental. The Armed Forces need to be smaller, not larger...