With a POD as early as, say, the 1850s, is it possible for the USA and the Second French Empire to go to war with one another in the mid/late 1860s? (Over the Mexican situation, perhaps?) If so, what would the outcome of such a war be?
That didn't stop them from invading Cochinchine, Madagascar or, as you mentioned Mexico. There was a sustained war ran for several years in Mexico, despite the fact than diseases meant you needed replacement often.No sure about whether it would be possible for them to go to war, mostly because neither is in a position to directly attack the other. Certainly, there's no way you could have an American invasion of France at that time, and France has no nearby colonies from which to base an invasion of the US.
For me the PODE is a Imperialist win at Puebla. If Maxie had sidelined Juarez and his Nationalists, I can see him saying to the Mexicans, "Follow me and my victorious army north of the Rio Bravo to take back our lost territory." Though, if I were him I'd settle for taking back a good chunk of Arizona and New Mexico. Texas and California would be too big a bites.
If for whatever reason France and the US get into a war, I would wonder how long Britain or Germany/Prussia might try to nip off parts of France or French Colonies.