Short answers, pulled from the story.
The Roman province of Mesopotamia was first established in early 116 by the emperor Trajan. He had conquered northern Mesopotamia by the end of 115 and organized it formally as a province. His successor Hadrian abandoned the province in 117, returning the Euphrates to its role as Rome's eastern boundary.
Septimius Severus re-established the province of Mesopotamia in 198, following his sack of the Parthian capital Ctesiphon. He deliberately modeled his action on Trajan's original conquest, naming Nisibis as the new provincial capital and raising it to the rank of a full colonia.
Under Septimius Severus, the province of Mesopotamia was bounded by the province of Osroene to the south, the Euphrates and Tigris to the north, and the river Chaboras, known today as the Khabur, to the east. It was smaller than the province Trajan had originally created.
The Persian shah Shapur I captured the Roman emperor Valerian at the city of Edessa in 260, during fighting across the province of Mesopotamia. Odaenathus of Palmyra subsequently engaged Shapur and forced the Persians to retreat.
After the debacle of Julian's Persian expedition in 363, Rome surrendered Nisibis, Singara, and territory in Adiabene. The provincial capital moved to Amida, and the military commander, the dux Mesopotamiae, relocated to Constantina.
Muslim armies swept through Mesopotamia between 633 and 640, permanently ending both Roman and Sassanid authority in the region. The East Romans had briefly recovered Dara after the war of 602-628, but the Muslim conquests ended their presence entirely.