Great discussion, everyone. Re: Napoleon III's ability, to my understanding he wasn't totally incompetent - indeed he seems to have been a pretty capable domestic ruler who greatly modernized the economy, expanded the French rail network and renovated Paris. It's true that his regime was in danger at times before 1870, but he navigated those problems by reforming the early absolutist Bonapartist monarchy into a parliamentary, semi-constitutional one. This way, he was able to retain popularity (as proven by his 1870 plebiscite where his reforms were approved by over 80% of the population) and even co-opt figures like Emile Ollivier (his last prime minister) who had previously been opposed to him, leaving the 'irreconcilable' republican types like Jules Ferry alone in the cold.
Of course, foreign affairs was a totally different story where the Emperor, to put it charitably, had a reverse Midas touch. But by 1870 the lesser Nappy wasn't long for the world anyway, as his health worsened through the 1860s to the point where he could barely ride a horse and he constantly suffered from gallstones, and he died from the latter in 1873. Foreign policy in the empire would've been directed not just by him but also by his wife Eugenie, an anti-Prussian ultramontane with very traditional views on how a monarch should behave (she was reportedly outraged when her husband surrendered at Sedan instead of fighting to the death IRL) who had been one of the foremost advocates for a 'short victorious war' to shore up the Bonaparte regime's popularity and who he appointed as his regent when he made his ill-advised decision to lead the French army directly. Since, as discussed, it seems very unlikely that France would be able to score an absolutely crushing victory over Prussia and doing so would open a can of worms the French would absolutely prefer remain closed, I think Eugenie would've been satisfied with the Saarland & securing the survival of the Bonapartist throne with her actually short and victorious war.
Once Napoleon III is dead and 1874 comes, the Prince Imperial will have attained his majority and been able to rule personally. From what I've read, Napoleon IV was cut more from his greatuncle's cloth than his father's, being a highly charismatic and gallant - if also hot-headed and rash - soldier who was historically well-liked by the British court (where he'd taken refuge after the fall of the 2nd Empire), made friends easily and was popular with the ladies. So long as he avoids Zulu assegais ITL, and with such a personality & the popularity he'd likely enjoy between his natural charisma & the successful FPW, I don't think he'd have any problem maintaining the Second Empire unless he loses an alt-WW1. Whether his youthful recklessness would translate to a Wilhelm II-esque propensity to alienate everyone around him as he gets older is an open question, it's not like he lived long enough IRL for us to know what he'd have been like as a ruler of anything, but personally I don't think it's too likely; at the least, Nappy IV doesn't seem to have any physical or psychological issues that would give him as horrible an inferiority complex as Kaiser Bill had IOTL.
As for foreign alignments for a Franco-Prussian rematch that snowballs into alt-WW1, I think that would also depend on when such a rematch happens. If it takes place in the 1870s or early 1880s, I think the Russo-Prussian alliance (with Italy as a third leg, possibly Britain too if Eugenie & Napoleon IV have been extremely reckless) battling a Franco-Austrian team is likely indeed. Unless the British are sufficiently alarmed at the prospect of the Russians & Prussians becoming masters of the continent to help the latter, I think that would end in a Russo-Prusso-Italian victory, perhaps even a quick one at that; the French army got some much-needed reform & modernization in the years following the FPW, which built on the reforms it was trying to adopt immediately before that disaster, but it can't possibly defeat all those opponents alone, and Austria-Hungary's army ( as well as those of the South German states) really doesn't seem like it'd be up to the task of fighting Prussia and Russia simultaneously at any point in the 19th-20th centuries.
However outside of the maximally-victorious scenario where they carve out 'Rhenania' either as a client kingdom or for themselves directly, I don't see a lasting Second French Empire being quite as powerful and menacing as the German Empire could be and was historically, so it's probably not enough to get Britain and Russia to set aside their differences - British support for the Turks, Russian ambitions against said Turks, and the Great Game the two had going - like they did IOTL against Germany, particularly if Alexander II still dies and is succeeded by the firmly anti-German & pro-French Alexander III. So to elaborate on my earlier thoughts, IMO it's more likely that Russia realigns toward France (and grudgingly Austria) for a war that kicks off anywhere between the late 1880s and 1914, while Prussia would pick up Britain (which I don't think would take the possibility for a resurgent Bonaparte hegemony in the west and Russian dominance in the east well at all) and the Ottomans to compensate.
As an aside, in the event of such a war I think the US would be inclined to neutrality at least at first, much as was the case historically. But I really don't see the Imperial French Navy being able to defeat both the Royal Navy above the seas and Prussian U-boats beneath it, so they wouldn't even have the ability to provoke the US with a blockade of the European continent (and Maximilian's Mexico has already fallen by 1870 so they don't have a ground presence to threaten America either) - while the 'Allies' of this alt-WW1 certainly could. If the Americans elect a sufficiently aggressive president, whether it's TR or someone else, could they be inclined to join the war on the French side over the Anglo-Prussian threat to their merchant shipping and go for Canada & the Caribbean? (Would be a way to get the Pacific War a couple decades early too I suppose, if Japan remains friendly with Britain and hostile to Russia as they were historically between the mid-19th century and WW1)