Lots of interesting points here. Now, LK and LD, this is my reasoned opinion about your arguments:
Regardless of how well Italy performs in the 1866 war, the Italian ruling elites are going to be rightfully terrified of fighting France without Prussia's backing, and would go out of their way to avoid it. So the main issue becomes whether ITTL Bismarck would be willing to fight by the time the Luxemburg and Rome crises arise, or not. I am qute skeptical that butterflies arising from a good Italian performance in 1866 would drive Napoleon and Garibaldi (whom the Italian government had little political control upon) to delay their bids on Luxemburg and Rome, if anything quite the contrary. Would the Italian government use police force to stop Garibaldi's organization of his ragtag volunteer army ? IOTL, they did not dare, since he was terribly popular as the only Italian general to reap victories against Austria; ITTL his victories would be rather less outstanding in comparison. I dunno if this would give the Italian government enough political capital to nip Garibaldi's attempt in the bud without excessive backlash. This is more or less the only way he can be kept available as a proxy against the Pope in later years. Otherwise, if Italy doesn't yet have Prussian backing for a war against France, they would do as OTL, give Garibaldi a free hand but no support behind a screen of plausible deniability and hang him to dry when the French defeat him. If Garibaldi fails in 1867, after previous defeats in 1849 and 1862, he's not going to try again in 1870, and liberation of Rome would hence by done at the hands of Italian regular army once the war starts, as IOTL.
This means that either OvB chooses to fight in 1867, or Luxemburg likely unfolds as IOTL, and the Mentana expedition is either delayed for 3 years by political butterflies, and hence may still be the casus belli, or happens just like OTL. In the latter case, if war occurs in 1870, the Spanish succession issue is still going to be the most likely casus belli. Now, the main candidates for the Spanish throne were an Hohenzollern and a Savoia, and with a strong Prussian-Italian alliance, both candidatures are going to be equally unacceptable to Napoleon III, and Bismarck is still likely to be able to make France freak out and become the aggressor in either case, just as it would happen with Luxemburg and Rome.
About the main issue, whether OvB would fight in 1867, my reasoned guess is that if Italy got the 1861 PoD and hence seized a full victory in 1866, he would dare to fight in 1867, the benefits of fighting France with such a strong ally seem to outweigh the risks of not having fully secured the diplomatic backing of Russia and the South German states. He would probably have to rush the effort to secure such a backing, but since both Russia and the South German states were already leaning towards Prussia (and would do more so if Austria looks even weaker than OTL in 1866), it does not seem to be too difficult.
Conversely, if Italy got the 1866 PoD and hence made a decent but not stellar performance in 1866, it is more likely that he would prefer to defer the war till he has secured all the points of his diplomatic network to secure the support of Russia and the South German states, like LK argued.
However, ITTL I do not really think that a pro-French faction in the Italian ruling elite is ever going to be a significant factor, unlike OTL: the king and the 'Piedmontese' faction were not idiots, if the Prussian-Italian alliance has worked quite well in 1866, and OvB hence confirms it, they are not going to switch alliances for the heck of it, even more so since France has turned hostile to Italy about the Roman question.
As it concerns the fate of Austria-Hungary, LK, I was not arguing for the creation of a Quadruple Monarchy, and the elevation of the Germans, Hungarians, Czechs, and Croats, to an equal level of confederal autonomy. While theoretically possible and it would ensure the Habsburg empire the highest degree of long-term stabilization, I deem such an outcome the by far least likely one of the Austrian political crisis, because it would require a degree of statemanship, foresight, and political courage that was sorely lacking in the Habsburg leadership. Rather, I was arguing that a slightly modified form of the OTL compromise might occur, full confederal autonomy to Hungary, a limited degree of subordinate federal autonomy to Croatia, and its ATL extension to Czechia. This would appease the Czechs to a limited degree like the Croats IOTL, and gain a measure of opportunistic support from them for the A-H status quo, if the Habsburg grow to deem Pan-German nationalism a significant threat ITTL.
However, ITTL I do regard the dissolution of the Habsburg empire as the most likely outcome, closely followed by an Ausgleich compromise even less stable than IOTL. Another possible but even less probable and stable outcome would be the return to the 1849-60 model after the constitutional attempts of 1860-65, a reactionary absolutist centralized monarchy, propped by conservative Habsburg loyalists and the opportunistic support of the Czechs and Croats to counter German and Magyar nationalism. However it would be a regime strongly opposed by liberals, German nationalists, and Magyar nationalists, with Habsburg loyalism much more discredited than in 1849 by 1859-66 military defeats, and so it would be very instable. The Quadruple Monarchy is the least likely possible outcome of all.
On the other hand, LD, I deem the scenario of an independent Hungary turning revanchist-expansionist and hostile to the German-Italian bloc as quite unlikely. In the mid-late 19th century, the Magyar ruling class consistently showed a deep committment to the preservation of their hegemony in the traditional borders of the Kingdom of Hungary, and, if at all possible, in the union with Croatia as well. Conversely they showed very little ambition to expand that hegemony beyond those borders and limited loyalism to the Habsburg, except insofar as the empire was opportunistically seen as a prop to support that hegemony. If Hungary becomes independent, it is almost sure to identify PanSlav and Romanian irredentism as its main enemy, and seek the support of those great powers that may support Hungary against it. Greater Germany and Greater Italy, as they would inevitably form ITTL alongside from the dissolution of the Habsburg empire, would likewise oppose Slav irredentism in their own territories, so a stable alliance is quite likely on this basis between Germany, Italy, and Hungary. Also because of this support, Hungary is almost sure to keep control of its minorities, and quite likely Croatia as well. Likewise, an independent Hungary is very very unlikely ever to show any kind of revanchism about Cisleithania. An hostility between Germany-Italy and Hungary is realistically only going to happen if Budapest seeks the patronage of Russia to prop up its own little empire, and Germany-Italy and Russia turn antagonistic, but since Russia is also quite likely to play the PanSlav card against the Ottomans at some point, this is much less likely to happen.
As it concerns the Pope, I would assume that if Rome is liberated by Garibaldi, the Pope is most likely to react as he did in 1848-49 and flee Rome, whileas if the Italian army liberates the Papal states, he is most likely to do as IOTL and hole up in the Vatican.
As it concerns France, I deem a Communard republic to be kinda ASBish, since, even if the French regular forces fail to suppress the insurrection by various feasible butterflies, German-Italian occupation troops are still going to be deep within French territory, on the outskirts of all or most of the revolutionary strongholds in fact, and I can see no valid political reason why Bismarck and the Italian government would let a far left revolution triumph on their borders and become a dangerous example and nest of subversion for their own peoples. Quite differently from an European intervention to crush the 1789 or 1917 Revolutions, ITTL the Italo-German repression of a Commune revolution in France would take relatively little military effort. But I agree that France shall always be revanchist, even more so than OTL since it suffered humiliation at the hands of, and lost territory to, Germany and Italy alike. But I agree that in all likelihood it is going to become either a reactionary kingdom or a republic, and the from of government is going to shape its behavior.
LD, the German and Italian navies in the 1860s-1870s were trivial in comparison to the Royal Navy, no sensible British leader is ever going to deem them a serious threat. TTL 1866-1870 victories simply switched the places of Germany & France and of Italy & Austria in the European pecking order, and made the Berlin-Rome duo the new main European land power in the place of Russia, after Napoleon and before the Crimean War, and France, after the CW. It is no ground to go into senseless paranoia, since the new wunderkind duo has much less ability to threaten British interests than either France or Russia, nor is their rise a seeming deadly threat to the balance of power. Britain in 1870 is strongly focused on building up and managing its empire, largely uninterested in continental affairs unless a bid for hegemony of Napoleonic latitude arises or Russia seems about to achieve a substantial advantage in its ongoing strategic rivalry with the British Empire in the Balkans, Middle East, and Central Asia. British attitude towards the Berlin-Rome duo is largely going to be shaped by their attitude towards Russia, friendly if they support its containment, hostile if they support its expansion.
Bismarck would never do, or allow his Italian sidekick to do (nor would the liberal Italian ruling elite have the megalomania to defy their main ally), anything that would give London a plausible reason to turn paranoid, i.e. threaten British interests in the Middle East. German-Italian seizure of Suez Canal shares is going to raise a few eyebrows in London, but the British allowed French-Egyptian control of the Canal without freaking out, and the French naval power was a much more credible potential threat to UK interests than German-Italian one. If anything, you may expect a diplomatic charm offensive by Bismarck to soothe UK nerves about Suez; an Anglo-German-Italian agreement to redistribute the Suez Canal shares on an equal basis between the three powers is quite possible, before or after the financial collapse of Egypt, and so is even an Italo-German-British protectorate of Egypt. Anyway, Britain has no reason to weaken its own share of control over Suez by giving part of it to France.