In 1867, Emperor Franz signed the Ausgleich, which officially turned Austria into Austria-Hungary. Both countries would have their own governments, with a shared head of state. Count Gyula Andrassy was the first Prime Minister of Hungary after 1867.
Helena frequently refers to Sisi as a peasant. In fact, Sisi was the grand-daughter of the King of Bavaria and her mother Ludovika was actually the sister of Archduchess Sophie of Austria (Empress Sophia in the series). Therefore Sisi was the cousin of Franz and of much higher rank than the Viscountess Helena.
In actuality, Sisi's relationship with Franz Joseph wasn't as romantic as in the show. While Franz Joseph was passionately in love with her, Sisi never felt fully the same way and was emotionally distant from him and she ended up avoiding him and court life as much as she could.
Sisi's sister Helene, who was likely the inspiration for the character of Helena, did eventually find a husband following rejection by Franz Joseph. She married the Prince of Thrun and Taxis and arguably had the only happy marriage of any of her sisters.
The inspiration for Chancellor Zottornick is possibly the Austrian Chancellor Klemens von Metternich. Their was a sour relationship between Metternich and the Archduchess Sophie, however she employed him to oversee the education of her son Franz in state matters.
Sisi was one of ten children of Ludovika and Max, however none of her siblings in the show matched her real siblings. Sisi was the fourth child and second daughter. She had five brothers and four sisters.
It is possible that the inspiration for the character of Helena von Grossberg came from the real life Sisi's own sister, Helene von Wittelsbach. Their mother Ludovika had originally intended for the Emperor Franz to marry Helene, but upon meeting both Helene and Sisi he fell in love with Sisi and insisted upon marrying her instead.