Short answers, pulled from the story.
Miklós Horthy took the oath as Regent of Hungary on the 1st of March 1920. He was the last commanding admiral of the Austro-Hungarian Navy and installed an authoritarian political system relying on traditional economic elites.
King Charles IV was arrested and exiled to Madeira because his attempts to return to power failed when much of the Royal Hungarian Army remained loyal to Horthy. The Diet passed a law nullifying the Pragmatic Sanction of 1713 on the 6th of November 1921, which dethroned him and abolished the House of Habsburg's rights to the throne.
The numerus clausus law formally placed limits on minority students at university and acted largely to conceal anti-Jewish action from foreign observers. Limitations were relaxed in 1928 when racial criteria were removed and replaced by social criteria favoring civil servants, war veterans, small landowners, industrialists, and merchant classes.
Land losses under the Treaty of Trianon caused Hungary to lose agricultural and industrial areas, resulting in budget deficits and high inflation. The country became dependent on exporting products from its remaining agricultural land until Prime Minister István Bethlen achieved monetary stabilization through large foreign loans and introduced the pengő currency in 1927.
The regency ended on the 1st of February 1946 when it was replaced by the Second Hungarian Republic. A High National Council had been appointed as the collective head of state before this date following the fall of the Szálasi regime and the German occupation of Budapest.