Short answers, pulled from the story.
Francis II created the title Emperor of Austria to protect his dynasty's imperial status. He foresaw either the collapse of the Holy Roman Empire or Napoleon becoming Holy Roman Emperor, and wanted to guarantee Habsburg power regardless of what happened to the old medieval order.
Austria fought Napoleon throughout most of the Napoleonic Wars, suffering defeats at Ulm, Austerlitz, and Wagram. It briefly allied with Napoleon during the 1812 invasion of Russia, then switched sides and played a decisive part in the 1813-14 campaigns that brought Napoleon down. Austria also participated in the second invasion of France in 1815.
Klemens von Metternich served as Austria's Foreign Minister from 1809 and as Chancellor of State from 1821 to 1848. He was the chief architect of the Congress of Vienna and steered Austrian and European foreign policy for over three decades. His belief in absolute monarchy and his elaborate system of congresses, spy networks, and censorship dominated the continent until the Revolutions of 1848 forced his resignation.
Hungary was legally a separate realm, described in its own 1790 constitution as a Regnum Independens. It was governed by its own King and Diet rather than by imperial institutions in Vienna, and had never been part of the Holy Roman Empire. This autonomous status was preserved throughout the empire's existence and ultimately led to the Austro-Hungarian Compromise of 1867.
The Bach System refers to the neo-absolutist policies of Baron Alexander von Bach, who dominated Austrian governance after 1852. It involved strict centralization of authority, restrictions on the press and public trials, and a concordat giving the Catholic Church control over education and family life. Critics described its four pillars as a standing army, a bureaucratic army, an army of priests, and a network of informers.
Austria's defeat in the Austro-Prussian War of 1866 fatally weakened its position. The German Confederation dissolved, Austrian influence over German-speaking lands ended, and Hungary pressed successfully for constitutional equality. The Austro-Hungarian Compromise of 1867 transformed the empire into the Dual Monarchy of Austria-Hungary, with Hungary and Austria joined as equal and separate entities.