What is it -- right now (or 2016,if you prefer) -- about America that isn't great?
I'm just going to give one thing that I think is probably the root of all the other issues: people are increasingly no longer able to have rational political discussions without one side or the other resorting to ad hominem attacks and then retreating to their information silos/echo chambers thereby further exacerbating the polarization of the public. People are often shunned just because they support a conservative idea or a marxist idea.
When was this time that the word 'again' implies? When was it exactly that those who endorse this slogan believe that America "was great"?
I think this question varies depending on who you're talking to, but for me, it's a combination of periods of time...I like the optimism of the Kennedy era, the economy of the Eisenhower or Clinton era, the strength of the Reagan era and the governmental libertarianism of the Washington era (without all the slavery).
What does a "great" America look like? Is it a reference to the fifties? Eighties? Pre 911? Pre political correctness?
This question is basically the same as the above. But yes, pre-political correctness is a good start. You can't have a frank and rational discussion about complex ideas without the possibility that someone will have their sensibilities offended.
Why exactly do they think that America "was great" then, but not now?
I think this question is answered above.
What values and beliefs are implicit in accepting this slogan and everything that goes with it?
The free exchange of ideas, judging people on the content of their character (meritocracy/MLK view) rather than their group identity.
Correspondingly, what values and beliefs does accepting this slogan require a person to thereby reject?
I feel it can best be summed up as a governmental policy of, "Clean your room before you try to fix the world." America first doesn't mean we don't help others, but if helping others comes at the cost of American citizens, then you should still default to America first.