Short answers, pulled from the story.
The Carolingian Empire began in 800 when Pope Leo III crowned Charlemagne as Roman Emperor, and it effectively ended in 887-888 when Charles the Fat was deposed and died. The empire lasted roughly 87 years in its unified form, though Carolingian rulers continued to hold power in the eastern and western kingdoms for decades afterward.
Pope Leo III crowned Charlemagne in exchange for political protection, deliberately disregarding the universalist claims of the weakened Byzantine Empire. The coronation took place on Christmas Day 800 in Rome.
The population of the Carolingian Empire is estimated at roughly 10 to 20 million people, based on calculations of empire size and theoretical population densities. Historian Timothy Newfield has challenged the idea of steady demographic expansion during this period, arguing that scholars underestimate the frequency of famines in Carolingian Europe.
The Treaty of Verdun, signed in 843, formally divided the Carolingian Empire among the three surviving sons of Louis the Pious. Lothar received the Central Frankish Realm and the imperial title, Louis the German received the Eastern Frankish Realm east of the Rhine (the precursor to modern Germany), and Charles the Bald received the Western Frankish Realm. The treaty is considered a foundational moment in the creation of what became France and Germany.
The Missi Dominici were paired imperial inspectors sent out to enforce Carolingian authority throughout the empire. One inspector was a cleric and one was secular. They made four journeys a year, each lasting a month, and were responsible for communicating royal laws, judging cases, and occasionally raising armies. A reform in 802 made the office permanent rather than ad hoc.
The Carolingian Empire fractured through a combination of civil wars over succession, the splintering of territories among heirs, Viking raids from the north and west, and the deposition of Charles the Fat in 887 after he bought off Viking raiders from Paris rather than fighting them. The death of Charles the Fat in 888 left no strong claimant, and the nobility elected regional kings from outside the dynasty or from its illegitimate line.