Short answers, pulled from the story.
Charles Martel earned the surname Martel after his victory over a Muslim army crossing the Pyrenees in 732. This battle marked the beginning of absolute rule for him over virtually all continental Western Europe north of the Pyrenees.
King Charlemagne was crowned Roman emperor by Pope Leo III on Christmas Day 800. This act disregarded the universalist claims of the weakened Byzantine Empire and is considered to mark the beginning of the seventh and final age of the world by intellectuals such as Alcuin.
The Treaty of Verdun in 843 divided the Carolingian Empire between Louis the Pious' three sons into the Central Frankish Realm, the Eastern Frankish Realm, and the Western Frankish Realm. Lothar received the imperial title and territory between the Rhine and Rhone Rivers while Louis received lands east of the Rhine and Charles received lands west of the Rhone.
The Carolingian Empire ended when Charles the Fat was deposed by the Frankish nobility in 887 after uniting all kingdoms for the last time in 884. He died in 888 and the empire immediately fractured with regional kings elected from outside the dynasty.
Stuart Airlie suggests that there were over 150 palaces throughout the Carolingian World which provided the setting for court activity. No permanent capital city existed yet Charlemagne began building the Palace of Aachen in the 780s where the palace chapel was constructed in 796.