As to the issue of corruption, there are all kinds of such things. If one forgets about religion for the moment and just considers business, there are petty affairs, and massive ENRON type conspiracies.
The question itself is very broad. There are financial escapades, and sexual ones. There is also influence peddling. Often times what appears to be a scandal of the church is actually the church as a victim.
As has been stated in other ways, corruptions can be found all over the human spectrum, and in every religion. Pastors who sleep with other men’s spouses can be considered a type of corruption, it is certainly an abuse of position (similar to the risk in any profession, I wouldn’t want my doctor or lawyer sleeping with my wife any more than my pastor).
The difference in church types can be the opportunities, and a highly centralized organization can give rise to highly sophisticated schemes of corruption. Certainly governments have experienced that, as well as large corporations.
In the Roman Empire the church was considered an organ of the state, and therefore a lot of church expenses were met by the government, is accepting that money a form of corruption? Possibly. Gifts of property (such as the turning over of public buildings) began as early as Constantine.
It is a sure thing that the state used it’s own influence over the church to control or manipulate the populace. (That continued to be a problem all over Europe for hundreds of years.)
The Roman emperors, being the “First Citizens” had a role to play as guardians of the nation, and protectors of the people, possibly even as instruments of God’s will. At least that was the theory, especially when there were hopes of having one universal kingdom on earth with one faith in Christ.
It then was also public policy to have the entire populace included under one church, and that meant the state was involved in church disputes (trying to prevent schisms, and trying to forge reconciliations). Very clearly the state consider unity in Christ a source of strength (and ultimately good for everyone). When schism occurred, the state did in fact become weaker.
The fact that the state attempted to subordinate all synods to it’s will was a form of corruption, in my opinion. That was a distortion of the normal ecclesiology which in modern times is not much of a problem. The situation became much much worse under the Caliphs and Sultans, who used the church like a fertile field to be plowed.
The west had a different situation, the church itself grew more centralized over time, with power concentrated in the hands of the bishop of Rome and his aides. Centralized institutions offer great opportunities for corruption.
One of the most significant western problems that I recall was the sale of church offices, which has been classed as simony. It became a much more common practice in the late middle ages and so accepted that the revenue was anticipated as a means to balance budgets of the bishops of Rome.
This form of corruption sometimes allowed for and encouraged another: the occupation of more than one See (another violation of the ancient canons), with a dispensation from the Popes. Revenue accumulated very rapidly that way, the money lenders getting back handsome returns for the advancement of funds for this purpose. It was a trilateral complex where three parties always benefited, and was difficult to suppress. (Something like the Military-Industrial Complex Eisenhower warned Americans about, and is still a problem fifty years later.)
As far as I know, the eastern churches did not have the problem of a bishop holding multiple Sees, because there was no mechanism to allow for it. Theoretically as well, monks couldn’t use the money, but I suppose there can be other motivations, like the advancement of family.
The bishops of the east for hundreds of years were mostly monks (the overwhelmingly preferred source for bishops), and they were watched by the state which would not hesitate to depose them, so I think sexual scandals were very rare. In any case, I don’t remember reading about them in the east.
Indulgences (of the type known in the west) were not sold, because there is no theory of satisfaction, no theory of the Treasury of Merits and no theory of Purgatory. Sincere confession being deemed sufficient to redeem a sinner.