Many things determine this, clearly. And one would think, sometimes, nothing more than dumb chance, based on which is first used and published and broadcast. Both linguistic and political factors might be at play. And sometimes even, poetic meter, I would argue.

Seriously, to take your first example "Spanish-American War", this arrangement produces a nice dactylic trimeter catalectic ("**\ **\ *") (for an example of dactyllic verse catalectic, see Browning's The Lost Leader). The alternative, "American-Spanish War" has no appealing flow to it at all.

The one rolls off the tongue with a rhythmic flow, while the other does not. As "French and Indian War" rolls off the tongue, using that (iambic) structure, but not using the hyphenated approach, and thus, "French-Indian War".

English has a strong preference (not obligate, but speakers find any opportunity to allow for it) for vowel reduction of alternating syllables. Sometimes, proper names (like "America") make this impossible, and the best we can do is reduce every third (as in "Spanish-American War"). But "French and Indian War" is much more our (iambic) preference.

And "which sounds better" will literally have been precisely the question addressed, in the era of mass media, covering war. In prior eras, which is more hospitable to the tongue nonetheless will have played a role.