Are there potential PODs for an earlier Austro-Prussian War? How might such a war turn out?
Well one of the main causes behind the Austro-Prussian War was the question of control in the German Confederation after the Second Schleswig War. The Austrians were nominally in control of the Duchies of Schleswig, Holstein, and Lauenburg, but the Prussians, who had just finished a crash reorganization and modernization of their military, were the ones who had done the majority of the fighting. Also the APR happened at such a time that there was little chance of outside powers intervening - Russia resented the Austrians for turning their back on them in the Crimean War, and the Prussians had helped them put down the January Uprising; Britain had withdrawn her commitment to help Denmark due to a change in government and a focus on internal matters (the Reform Act of 1867); France was pure luck, as Napoléon III had had plans to intervene in the war on Austria's side but changed his mind the night before the invasion was to go forward. So you would need a similar test of strength between the two vying powers ITTL, without foreign interference, to lead to an earlier war. That's not very likely to happen.
The best I can think of is that Prussia rejects the Punctation of Olmütz, and Prussia and Austria go to war over the question of intervention in the Electorate of Hesse and in the Kingdom of Bavaria. Perhaps the POD is that von Manteuffel is over-ruled in the recently re-established United Landtag by hardliners. Frederick William, Elector of Hesse, under the influence of his Minister-President, Hans Daniel Ludwig Friedrich Hassenpflug, had seceded his domain from the Prussian-led Erfurt Union, and Hassenpflug himself traveled to Frankfurt to the recently re-established Diet of the German Confederation to ask for Austrians intervention after Prussian troops seized Kassel (and the Bavarian town of Fulda after Bavaria also withdraw from the union just days later). IOTL the Prussians backed down after Prince Felix of Schwarzenberg issued the Punctation demanding Prussia withdrew its forces under threat of war with Austria, with von Manteuffel declaring that "The strong man takes a step back." ITTL though if von Manteuffel is over-ruled, or Hassenpflug less successful, or both, then war would break-out between the two German powers. Prussia had just recently negotiated the Alliance of Three Kings with Hanover and Saxony; however in Saxony and Prussia itself radical democratic agitation would be high, and the Allies' troops would likely be just as busy putting down revolts and uprisings as they would fighting the enemy. On the other hand though the Hapsburgs would still be incredibly busy holding down the Italians and Hungarians, and if it came down to it Vienna would much more likely want to hold on to the Pannonian Basin and the North Italian Plain than they would their influence over the German states. Neither side is really prepared for, or wanting, a war right now though. And unlike IOTL, there's a strong possibility of Russian intervention; France is too busy with Napoléon III touring the country preparing for his coup d'état by securing the allegiance of the military and the people, and in Britain while Palmerston might press for immediate intervention the rest of the Whig government of Russell is unlikely to go along with such a move - which might, incidentally, lead to an earlier fall of the Russel government, in which case the IOTL Who? Who? Ministry is likely skipped right over, and a coalition of Palmerstonian Whigs and Peelite Conservatives comes to power, which would likely lead to British intervention in late 1851 or early 1852, depending on how the war is going and if the Russians have intervened by then or not.
The ultimate result of such a war is up in the air - like I said, neither side is really prepared for it. If the radicals become a big enough issue, especially in the smaller German states, the Austrians and Prussians might set aside their rivalry long enough to crush the democrats once again. Ultimately I think you'd come to see the German Confederation finally rendered moot, with a new *North German Confederation of Prussia, Hanover, Saxony, and any allied and conquered states against a looser alliance of southern German states under Hapsburg leadership and protection, with the threat of foreign intervention hanging over both.
That doesn't really solve the 'German Question' question though, for either the Germans or for Napoléon III, who is unlikely to be comfortable with German unification happening in such a matter (e.g., one that doesn't aid him).