Perditianus on Reddit pointed out on May 16 that this is exactly what Google Translate gives for “Live free or America dies”. So it seems likely that this piece of text was not composed in Latin by any human author.
I don’t think “What does this mean” is a clear question when applied to a sequence of words produced in this manner. If you consider its ...
As the other answers indicate, this is nonsense. But I think it would be helpful to provide (1) a parsing of the nonsense Latin, and (2) a good translation of the intended phrase.
Parsing of nonsense Latin
vivamus: 1st person plural subjunctive, "let us live"
vel: (inclusive) "or"
libero: this can either by the 1st person of libero ("I deliver/free") or ...
I think it's Google Translate nonsense, but it's perplexing that it'd find its way to a cover.
The results may depend on the user, but I get these translations:
Live Free or Die: America > Free aut mori; Americae
Live free or Die: America > Liberum vivere aut mori; Americae
Live Free or die: America > Free aut mori; Americae
Live free or die: ...
No, that translation is not grammatically valid.
It means roughly "belief of law, long use, to be saved firmly" but it is somewhat incoherent.
Let me go through a translation process step by step.
As you seem to know, opinio iuris is a fixed expression and we can of course start with that.
The modifiers will probably not be parts of standard legal Latin ...
Cedere takes the dative, so it should be:
Ne apathiae cede, sed humanitati.
(Liking this word order better, but yours is fine too, of course.)
Other than that, I see no objections. Apathia and humanitas are fine (no false friends); you could also say lentitudo for apathy.
I would change your title to "Potens (est) Imperator." This translates as "Powerful/Capable (is) (the) Emperor." I like this translation better because Latin prefers adjectives to genitives (as far as I can tell) and succinctness.
As Jasper has answered, "Per Imperium Venit Pax" is a good way of declining your motto.
I would analyse non obstante eo quod as “in spite of the fact that …, the fact that … notwithstanding” etc. (literally: “with it not hindering that …,” i.e., it is an ablativus absolutus). And why the subjunctive? I would say it is another case of the dreaded quod cum coniunctivo.
I think you got the first part right but started stumbling at siue denique, ...