Short answers, pulled from the story.
The Abbasid Caliphate began in 747 when Abu Muslim raised a black flag in Merv to start an armed uprising. This revolt toppled the Umayyad dynasty within three years by drawing strength from Khurasan.
Caliph al-Mansur laid the foundation for Baghdad on the banks of the Tigris River in 762. He named this new metropolis Madinat al-Salam or City of Peace and constructed successive rings of brick walls with a workforce of 100,000 laborers.
Turkish soldiers murdered Caliph al-Mutawakkil on the 1st of January 861 which marked the first direct military intervention at court. Four different caliphs came and went within nine years while tax collection lapsed and the state remained short on funds.
Mongol forces under Hulagu Khan besieged Baghdad starting on the 29th of January 1258 and began the sack on February 13. They slaughtered the population and trampled Caliph al-Musta'im to death by horses wrapped in a carpet on the 20th of February 1258 ending five centuries of rule.
In 1261 Mamluk ruler Baybars invited a surviving member of the Abbasid family to establish himself in Cairo as a new caliph. The line continued there as a strictly ceremonial institution conferring prestige upon the Mamluk sultans until Ottoman Sultan Selim I conquered Egypt in 1517.