I�m not a scholar, so take my answers with a grain of salt.
May I please have some explanation as to why the western monks/nuns became school teachers and eastern ones have not?
Well, first, it�s important to stress that Western monks were not primarily school teachers. They were primarily monks. They felt called by God to live apart from �the world� and to devote themselves to becoming holy (or, at least, righteous) and to help others do likewise.
However, monasteries in the West also became centers of learning because they were usually the only places where people could read and where there were books. That is because the barbarian invasions of Western Europe had destroyed the Western Roman Empire and they had pretty much wiped out the system and tools of education there. About the only consistent exception was the monks. So, it was natural that their monasteries would also become centers of education in a dark sea of illiteracy. This began to change around the year 1000, when Western Europe again became settled and civilized enough for secular education to be revived.
Meanwhile, the Eastern Roman Empire did not fall to barbarian invasions. As a result, the system of secular education there was preserved. Thus, Eastern Christian monks were free from the additional task of being teachers of secular knowledge (as their Western brothers were occupied). Instead, in the East, people could get a secular education from secular teachers; and monks could focus exclusively on the vocation of the monk: prayer, work, (religious) study and liturgy.
Is there or has their ever been even an eastern/oriental school system? if so who were the teachers it if not monks?
Yes. From what can be deduced from historical sources, education in the Byzantine Empire seems to have been (roughly) a two-tiered system.
The first tier of Byzantine education was by private tutor or teacher. In other words, a family would hire a teacher, or a family would send a child to the private school of a teacher. This first tier was a basic education: knowledge of the Greek language (grammar, literature, rhetoric), some math, a wide study of history, and a good study of geography.
The second tier of Byzantine education was professional education: law, medicine, engineering (military and civil), and (in the monasteries) theology. Professional education was imparted by members of those professions. Sometimes this was done at academies (a kind of precursor to the university) that were established by the government. Sometimes, it was done on the job. Sometimes, it was by private (i.e., tutored) instruction.
It should be noted that education was generally a quality of the wealthy classes (nobles and upper middle class) or government bureaucracies. In other words, there was no system of compulsory public education for everyone. Instead, the first tier of education (i.e., a basic education) was usually the responsibility of the family; the second tier (i.e., a professional education) was partially private and mostly founded and directed and subsidized by the state.
For this, I rely especially on the chapter on education in the slim but good book �Byzantine Civilization� by Steven Runciman. Again, I�m not a scholar; and there are probably more recently written treatments of this subject.
Did the Eastern Roman Empire continue to function much as the Western one did before it fell/declined?
That is a topic that is wide open for much discussion and debate. Here is my opinion; keep in mind that others can see things differently.
The Eastern Roman Empire functioned more or less as the Old Roman Empire from its founding by Constantine (330 A.D.) till the reign of Heraclius (610 - 641 A.D.). The major differences during this period of time were twofold. It was based in the East at Constantinople (instead of at Rome). Gradually, the Greek language became more commonly used than the Latin language (because more Greeks than Latins lived in the Eastern Roman Empire).
By the rein of Heraclius (610-641 A.D.), however, there was an enormous change: the rise of the Muslim Caliphate. The Muslim Caliphate permanently conquered about 3/4 of the territory (and people) of the Eastern Roman Empire. Before the rise of the Caliphate, the Eastern Roman Empire consisted of Greater Greece (for lack of a better term: the predominantly Greek speaking areas, including the Balkan peninsula and most of Asia Minor), North Africa, Egypt and Syria/Palestine. The Caliphate, however, conquered three of these four areas: North Africa, Egypt and Syria/Palestine.
Hence, there were three major changes in the Eastern Roman Empire because of the rise of the Caliphate.
1. It became more Greek.
As mentioned above, the Caliphate permanently conquered � of the Eastern Roman Empire. All that was left was the Greek portion of the empire. Hence, by default, the empire became more Greek. Starting with the emperor Heraclius, Greek officially replaced Latin as the language of the empire.
2. It became more feudal.
Before the reign of Heraclius, the imperial army was usually separate from the government of the provinces. Beginning with Heraclius, however, this changed. With the new �theme� system of government, the military became the administration (the government) of the provinces. Specifically, units of the military were permanently located in certain provinces, and those units governed their provinces. They did so in the name of (and under the control of) the central government. However, the units of the military obtained most of their support and personnel locally -- from the provinces in which they were located. This was accomplished by land-grants. Grants of land were given by the emperor in exchange for hereditary military service. This was another big change. Previously, grants of land were given by the emperor as a reward for an individual�s career in military service to the empire. Now, land was given as support for a career in military service. Essentially, this reflected a change in times. The empire was no longer expanding, and soldiers could no longer be recruited by the promise of owning their own freshly conquered lands. Instead, the empire was on the defensive, and soldiers were recruited by being given lands in their area which then would support them while they defended those lands. The result was adding a feudal element to Byzantine society: land in exchange for military service. Byzantine society did not become purely feudal. The empire was still based on a strong, centralized government; and the government was understood as a state. (This is in contrast to the feudal ideal, in which �government� was the person of the leader --the king, duke, etc.-- and whoever had personally sworn loyalty to that leader.) Nevertheless, the feudal element did have an effect on Byzantine society. Over time, local nobility (who owned most of the land and who comprised much of the military) tended to gain power at the expense of the central government.
3. It became a nation-state.
After the reign of Heraclius, the Eastern Roman Empire was no longer a multi-ethnic state that was dominated by the Roman people. Instead, it was mostly a Greek state: in language, religion and land. Furthermore, society was no longer based mostly on a civic idea of the state. Instead, it was more and more a mixed system of feudalism and the idea of the state. It still thought of itself as the Eastern Roman Empire, and there was a continuity in religion, law and culture. However, in my opinion, the Eastern Roman Empire ended during the reign of Heraclius and with rise of the Muslim Caliphate. In its place (in my opinion) was a new state: based upon the Greek remnant of the Eastern Roman Empire that had not been conquered by the Muslim Caliphate. It was the medieval Greek nation-state that is called the Byzantine Empire.
-- The above was my personal opinion; others have different opinions. The term �Byzantine Empire� traditionally refers to the state that had its origins in the Eastern Roman Empire and which perished with conquest by the Ottoman Turks. The traditional dates are 330 A.D. for the founding of the city of Constantinople and 1453 A.D. for the conquest of that same city by the Ottomans. There is quite a bit of difference of opinion about when the Eastern Roman Empire ended and when the Byzantine Empire began because there was so much continuity from the past despite its serious changes. In my personal opinion, I date the beginning of the Byzantine Empire to the reign of Heraclius for the reasons I stated above. Others, however, would date it differently.
-- A good introductory essay on the Byzantine Empire can be found at http://en.wikipedia.org/wiki/Byzantine_empire
-- A good introductory book is �Byzantine Civilization� by Steven Runciman.
-- A good full-sized textbook is �The Byzantine State� by George Ostrogorsky.
-- A good introduction to the religion of the Byzantine state (and, later, other peoples), the Orthodox Church, is �The Orthodox Church� by Bishop Timothy Kallistos Ware.