What is your evidence that either the Pope or the Ecumenical Patriarch (or any other patriarch) appointed any bishop before the schism? I'm not familiar with details of Orthodoxy, but a Web search suggests that even today, bishops may be elected by various means.

This is a broad question, especially because it lakes a time frame, but the short answer is yes there were conflicts relating to who had the right to appoint bishops. The process of who appoints a bishop has changed several times throughout history, as has their traditional roles. In early Christian times, Bishops were almost indistinguishable from Deacons, who were church elders elected by their congregations. By the time the persecutions of Christianity were banned by Constantine, the religion was extremely popular, especially the east. After winning the wars of the tetrarchy, Constantine oversaw a number of reforms in doctrine in an effort to bring stability to his new christian empire. He took a personal role in defining christian doctrine through the Nicene creed, and appointed Bishops to important posts within his empire. During the reign of Theodosius I, Christianity was made the state religion, and church officials gained even more power and influence.

At this time, the Emperor in Constantinople was considered the head of the church, and protector of Christendom. This situation persisted until around 800AD, when the Bishop of Rome Leo III crowned Charles I King of the Franks as "Emperor of the Romans". There are several theories why this occurred, but part of the rational was that the throne of Constantinople was occupied by the Empress Irene, and thus considered vacant to western churchmen. Irene was soon replaced by her former brother in law Nikophoros, and the two competing crowns for the "Emperor of the Romans" continued until 1054, when the Patriarch in Constantinople Excommunicated the Bishop of Rome, and visa versa. This is considered the political split of the Church into the modern forms of the Latin West, and Orthodox East, and after this time the two empires governed their ecumenical affairs separately. During this time, the Title of Emperor of the Romans had fallen out of the Carolingian family, and been vested in the German Duke Henry the Fowler, and his son Otto I. This prompted the question of who had the right to appoint Bishops. Following Charlemagne's example, Otto I and his successors had appointed Bishops in the Constantinian style. However, the Bishop of Rome with his new western primacy in church affairs, began to assert his right to appoint bishops instead of the Emperor. This struggle became known as the investiture controversy, and as a result the ecumenical power in the west shifted away from the Emperor, and toward the Bishop of Rome.

In the subsequent periods the fortunes of the Papacy rose and fell, to their lowest point in the 1300's during the Avignon papacy, when the popes were once again little more than puppets of the french king. They reached the high point in the later 15th early 16th century with the papacies of Alexander VI and Julius II. These two popes turned the papacy into the temporal and ecumenical powerhouse it is historically conceived of today, and the papal states continued to dominate the politics of Italy until Napoleons Invasions in 1798, and 1808. In the modern Latin Catholic church this tradition is reflected by the fact that all nominees for bishop are confirmed by the Pope, even if only formally.