The Catholic pope is also the bishop of Rome, since St Peter. However, the Papacy moved to Avignon from 1309 to 1376. Were these popes (seven in all) still considered bishops of Rome? Bishops are required to reside in their diocese, so how would this have been reconciled?
Yes. This is why St. Catherine of Siena urged Pope Gregory XI to return to Rome; a pope is supposed to be bishop of Rome, in Rome.
One of her 1376 letters to then-Avignon-abiding Pope Gregory XI (from Saint Catherine of Siena as Seen in Her Letters; Italian original):
Come [back to Rome] in security: trust you in Christ sweet Jesus: for, doing what you ought, God will be above you, and there will be no one who shall be against you. Up, father, like a man! For I tell you that you have no need to fear. You ought to come; come, then. Come gently, without any fear.
John N. Deely, Four Ages of Understanding: The First Postmodern Survey of Philosophy From Ancient Times to the Turn of the Twenty-first Century, pp. 396-7:
from this answer to the Christianity StackExchange question "In Catholic understanding when there is more than one Pope, who decides which Pope is real and by what criteria?":
Regarding this Great Western [Papal] "Schism",* when even saints disagreed about who the valid pope was,** Canon J. Didiot gives this analogy:
If after the election of a pope and before his death or resignation a new election takes place, it is null and schismatic; the one elected is not in the Apostolic Succession. This was seen at the beginning of what is called, somewhat incorrectly, the Great Schism of the West, which was only an apparent schism from a theological standpoint. If two elections take place simultaneously or nearly so, one according to laws previously passed and the other contrary to them, the apostolicity belongs to the pope legally chosen and not to the other, and though there be doubts, discussions, and cruel divisions on this point, as at the time of the so-called Western Schism, it is no less true, no less real that the apostolicity exists objectively in the true pope. What does it matter, in this objective relation, that it is not manifest to all and is not recognized by all till long after? A treasure is bequeathed to me, but I do not know whether it is in the chest A or in the casket B. Am I any less the possessor of this treasure?
*Technically, it was not a schism since all those involved were Catholic.
**From the "Western Schism" article: "St. Catherine of Siena, St. Catherine of Sweden, Bl. Peter of Aragon, Bl. Ursulina of Parma, Philippe d'Alencon, and Gerard de Groote were in the camp of Urban; St. Vincent Ferrer, Bl. Peter of Luxemburg, and St. Colette belonged to the party of Clement."