Short answers, pulled from the story.
The Qajar dynasty ruled Iran from 1789 until 1925, a total of 136 years. The dynasty was founded through the Unification of Iran by Agha Mohammad Khan Qajar between 1779 and 1796, and ended on the 12th of December 1925 when Iran's Majlis declared Reza Shah the new ruler.
Naser al-Din Shah Qajar held the throne for 47 years, from the 5th of September 1848 until the 1st of May 1896, making his the longest individual reign in the dynasty's history. The dynasty's second longest single reign belonged to Fath-Ali Shah Qajar, who ruled for 37 years.
The Qajar dynasty ended on the 12th of December 1925 when Iran's Majlis, sitting as a constituent assembly, declared Reza Shah the new shah of Iran. Reza Shah had previously been a brigadier-general of the Persian Cossack Brigade, and his accession began the Pahlavi dynasty.
Qajar shahs adopted titles including Shahanshah of Iran, Zell'ollah meaning Shadow of God on earth, and Qebleh-ye 'alam meaning Pivot of the Universe. These titles blended pre-Islamic, Islamic-Shi'i, and nomadic elements of Persian governance to assert divine authority and political legitimacy.
Sultan Mohammad Ali Mirza Qajar has headed the Qajar imperial family in exile since 2011. The headship passes to the eldest male descendant of Mohammad Ali Shah. The current Heir Presumptive to the Qajar throne is Mohammad Hassan Mirza II, grandson of Sultan Ahmad Shah's brother.
Abdol Hossein Sardari was a Qajar Qoyunlu and, through his mother, a grandson of Princess Malekzadeh Khanoum Ezzat od-Doleh, sister of Naser ed-Din Shah. Serving as Consul General at the Iranian Embassy in Paris from 1940 to 1945, he saved Jews facing deportation by issuing them Iranian passports.