Short answers, pulled from the story.
Operation Panzerfaust was a German military operation carried out in October 1944 to prevent Hungary from surrendering to the Soviet Union. Hitler feared that Hungary's defection would expose Germany's southern flank and cut off a million German troops fighting in the Balkans.
Otto Skorzeny, a Waffen-SS commando leader, directed the operation on the ground. He organized the kidnapping of Miklós Horthy Jr. and personally led the tank convoy to Castle Hill in Budapest to force Regent Horthy's capitulation.
Miklós Horthy Jr. was lured to a meeting at the offices of Felix Bornemisza, Director of the Hungarian Danube ports, on the 15th of October 1944, under the false pretense that envoys of Marshal Tito wanted to speak with him. Skorzeny and his troops attacked him, wrapped him in a carpet, drove him to the airport, flew him to Vienna, and sent him to Mauthausen concentration camp.
Horthy signed the document renouncing the armistice and abdicating in favor of Arrow Cross leader Ferenc Szálasi because he was told his son's life was at stake. He later described it as exchanging his signature for his son's life, stating that a signature extracted at machine-gun point carried little legality.
Horthy was transported to Schloss Hirschberg near Weilheim, Germany, and guarded by one hundred Waffen SS men. On the 1st of May 1945, US 7th Army commander Lieutenant General Alexander Patch visited him there. Horthy was released from the Nuremberg penitentiary on the 17th of December 1945 and reunited with his family in Weilheim.
Operation Margarethe was Germany's occupation of Hungary in March 1944, carried out to secure Hungary's continued participation in the Axis. It preceded Operation Panzerfaust and enabled the deportation of Hungarian Jews, who had previously been beyond the direct reach of Nazi policy.