Short answers, pulled from the story.
Archduke was a title of nobility in the Holy Roman Empire, ranking above duke but below king or emperor. It was invented by Rudolf IV, Duke of Austria, who fabricated the Privilegium Maius in 1358-1359 to claim the rank for his dynasty.
The title was officially recognized in 1453 by Emperor Frederick III, who awarded the archducal rank to the Austrian ruling House of Habsburg, elevating the Duchy of Austria to the Archduchy of Austria.
Albert VI of Austria, Frederick III's younger brother, was the first to use it in practice, doing so at least from 1458. Emperor Frederick III himself continued to use only the title Duke of Austria until his death in 1493.
Rudolf IV created the title to place Austria on equal footing with the Prince-Electors of the Holy Roman Empire, after Austria was excluded when the Golden Bull of 1356 assigned that dignity to the four highest-ranking secular princes and three Archbishops.
Official use of the Archduke title has been illegal in Austria since the Law on the Abolition of Nobility passed on the 3rd of April 1919. Habsburg family members residing in Austria are known by the surname Habsburg-Lothringen, while those in other countries may use the title according to local law.
Yes. Marie Antoinette was born Archduchess Maria Antonia of Austria. From the 16th century onward, the title Archduchess was used by all female members of the House of Habsburg.