A don't see why the Schleswig-Holstein question between the German states and Denmark would be a matter they would care enough about to get involved militarily
The simple reason was that Austria was part of the German Confederation and the Schleswig Wars were about protecting the rights of the German Confederation in these duchies. The German Confederation was a political organ set up in the aftermath of the Napoleonic Wars to recreate some aspects of the dissolved Holy Roman Empire. The Confederation included areas that were nominally "German" in the strictest ethno-national sense of the word. Areas like Bohemia or the Galician duchies had a strong German-speaking minority, but were included as components of the Confederation for geopolitical reasons rather than any sense of ethnic kinship. In Galicia's case, Metternich pushed for its inclusion in the Confederation as a means to guarantee the German states' involvement should Russia go to war with the Austrian Empire. Galicia was one of the main possible invasion routes for such a conflict and their inclusion meant a Russian attack there would legally be an attack on the Confederation, not simply a sovereign matter between Austria and Russia.
Metternich and other Austrian statesmen saw it was in Vienna's interest to use the Confederation to promote Austrian interests within Central Europe. In short, the German states were too important for the balance of power to be left to their own devices. Austrian statesmen recognized that Austria's absence in the Confederation would have led to the domination of Germany by Prussia. This was something that had happened in the eighteenth century when the successes of the Hohenzollerns allowed them to act selectively within the HRE's collective structures. Prussia was strong enough to ignore HRE strictures when it suited Prussian needs of state, such as the seizure of Silesia, but it also could act as a "protector" of the HRE structures too. Prussia was the leader of the anti-Austrian coalition in the so-called Kartoffelkrieg (Potato War) of 1778/9 which prevented the Habsburgs from gaining control of Bavaria. Metternich sought to redress this problem when crafting the Confederation by making Austria an integral and dominant member of the Confederation. And this did work in a manner of speaking. Austria was able to exert influence over German affairs for the fifty years after Napoleon's defeat. The Olmützer Punktation in the wake of the failed 1848 Revolution in which the Frankfurt Paulskirche parliament tried to unify Germany without Austria led to a strengthened role of Austria in the Confederation.
But Austria always had one foot in German affairs and another within Europe which made it vulnerable to Prussian attempts to undermine the Confederation. The wars over the Danish duchies put Austria into a bind. As one of the two strongest members of the Confederation, Austria had to intervene in the interests of Germany lest Prussian statesmen accuse Austria of abandoning Germany. But the control over the Duchies gave Vienna one additional care in an empire that was overstretched. The Habsburgs had to assimilate Danish affairs in addition to their Czech, Hungarian, Polish, and Italian concerns. This was a chink in Austria's geopolitic armor that Bismarck would exploit in the subsequent Austro-Prussian War