The short answer: there is no substantial evidence that the elite did not "believe in God". There is evidence that at least some elites did not particularly care about the specifics of organized religion, or were otherwise flexible in their beliefs—but these reveal early modern sensibilities rather than an underlying skepticism about the existence of God or of the basic tenets of the Christian faith. In this answer, I am primarily going to focus on the political elite in Germany and England. I will begin with a general statement, and then get into the specifics of the question.

To the early modern European, even the elite, a question on the existence of God just would not make sense. Everything early modern Europeans read, saw, created, and wrote accepted the existence of God—"atheism", as we know it, was considered the provenance of madness or utter depravity. Without the base of modern empirical science to look at, explaining the functioning of the world without, at least, the existence of at least some "supernatural" forces would seem necessary and undeniable. Famously, even leading geniuses like Isaac Newton, who one might expect to be leading skeptics of faith, spent more time writing about religion and alchemy than about what we would today call "science". Religion was a part of every European’s life, from birth until death.

A better question might be, "how sincerely did elites believe in the specific forms of the Christian faith practiced in the early modern period?" The answer is not provable, of course, but an attempt to answer can still tell us much about early modern elites and their relationship to religion and society. I’ll give some examples after I run through your specific statements a bit.

Literacy rates at the time were particularly low in the poorer classes which meant that they were unable to read the Bible independently..

The Church undoubtedly maintained dominance over the domain of religion (although we should not dismiss the importance of local piety). However, church officials were by no means a monolithic entity and the vigor of their internal debates over theology and dogma both before and after the Reformation—debates which would completely avoid the attention of even most other literate people—suggests that they considered questions of faith and religion to be serious. Debates over Scholasticism and its specific occupied centuries of scholars, for instance. Discussion about the morality of so-called magic, like alchemy and astrology, were prominent in Renaissance Europe, as elites explored the supernatural powers of man and debated from whence such powers came. Richard Kieckhefer’s Magic in the Middle Ages is a wonderful overview of the complexity and seriousness of such debates. Keep in mind, additionally, that the Church (both Catholic and Protestant) itself funded the vast majority of education in the medieval and early modern period.

You are not wrong about the importance of what we now would call “personal relationships” with God, or at least Jesus, in Protestant theology. Translation of the Bible into the vernacular and the demotion of the priest from the position of mediator between God and layman were (and are) central planks of Protestantism. Particularly in the Anglophone world, which Protestantism has heavily dominated since the English Reformation, we are generally taught to see these beliefs as incredible improvements over what Catholicism offered, and popular accounts often pitch Protestant ideas as amenable to what the lower classes actually wanted. However, this is a grave misunderstanding of how the early modern world appeared in the perspective of most Europeans.

The early modern peasant often had a surprising reaction to Protestantism. After Martin Luther decisively distanced himself from the cause of the peasantry during the Peasant Revolts, the vast majority of the lay populace appears to have been, at best, ambivalent to Protestantism. On the local level, Catholicism offered a great deal to the peasantry: the parish church was a symbol of local pride and prosperity, while veneration of saints brought a personal touch to worship. The Eucharist is, essentially, a form of magic that their parish priest performed before them at Mass, and the Latinate liturgy engendered the idea that Mass was a special space where the parishioner could feel as though they had stepped into another world (on their annual appearances at Mass, at least). Eamon Duffy’s engrossing Voices of Morebath shows how an average English parish saw the Reformation. A lot of the non-elite liked Catholicism, and many were even willing to raise up arms in order to bring back the Pope's rule, the monasteries, and the deep mysteries and complexity of Catholic worship.

This has been a bit of an aside, however, because you want to know about the elites.

Does anyone have any information on why exactly the Catholic Church was seen as corrupt?

Simply, because the Church was corrupt. We have to understand, however, that “corruption” was a fact of life in early modern Europe, rather than an aberration practiced only by churchmen and somehow illustrating a lack of sincerity. Most “corruption” in the early modern period centered primarily around patron-client relationships, which were mutually beneficial arrangements between people of different social strata. Giving and receiving favors, positions, and wealth in exchange for the same was simply how early modern Europe functioned; it may seem unconscionable to the modern mind, but the idea of meritocracy had little traction in most of European society. Who you knew, and what who you knew could do for you, was more important than your talent or ability. Protestantism would only very slowly shake up this system, since it merely removed the source of “corruption” from the Catholic Church to the State. Paul D. McLean’s The Art of the Network is an accessible work on patronage in Renaissance Florence.

The fact is that during these times the churches paid little tax and held incredible wealth. Combined with the fact we know that the popes of the times were not as pious as they claimed, iirc one of them contracted syphilis, complements the theory that they were pretending to be more religious than they were.

All this suggests is that such popes (although their alleged immoralities appear to have been overstated) were less moral than they may have claimed to be. Hypocrisy itself is not a sign of lack of belief—a smoker, for instance, can tell you not to smoke cigarettes, and the fact of their smoking does not negate their apparent belief that smoking is unhealthy or wrong.

The divine right of Kings relied on the fact the king was a pious man, so to assure a strong powerbase it was somewhat necessary to say you were religious. Henry of Navarre converted to Catholicism in ?1594? due to political pressure so this is an example of using religion for political gain.

Here’s the real meat: we cannot separate “religion” from “politics” in the early modern period. Henry of Navarre is generally claimed to have said that being King of France was “worth a Mass”, which suggests that he was just cynical (and that certainly appears to be the case); on the other hand, Henry’s flexibility protected France from the continuation of a brutal religious civil war that threatened the very existence of Protestantism in France. His interest in power is very clear, but no monarch would deny that they ruled “Dei Gratia”—that is, by the grace of God. Religious and political motivations overlapped inextricably.

This overlap existed across Reformation Europe, resulting in wars where religious identity enmeshed so thoroughly with politics that princes and nobles were willing to risk everything to protect and expand their faith. Calvinist Frederick V, Elector Palatine, cast the dice that brought about the Thirty Years War when he attempted to take the crown of Bohemia in order to protect the rights of the Protestant nobility there (and, maybe, bring about the coming apocalypse in which righteousness would finally end Catholicism). Queen Elizabeth I of England famously helped the Dutch rebels against the Spanish (which resulted in the famous defeat of the Spanish Armada), even though she rejected their offer to be their Queen repeatedly. Lutheran King Gustavus Adolphus of Sweden died on the field of battle in Germany, in battle against the Catholic Habsburgs; his Swedish troops conquered some of the Empire, but also went far afield to stop the re-Catholicization of the defeated Protestants. There are examples of much more mercenary choices, of course, like Catholic France’s attack on the Habsburgs at the end of the Thirty Years War, but these choices reflect the large variety of political and religious considerations undertaken by princes and the elite throughout the period.

Like all of humanity, every member of early modern elite was capable of piety and sacrilege, sincerity and opportunism, avarice and charity—many elites managed to represent all of these at different times. There is no evidence of a massive conspiracy to keep the masses pious, and plenty of evidence to the contrary, whether in elite willingness to go to war with each other over issues of faith, to scholarly debates over the nature of faith and religion, to the apparently sincere religious practice and attendance of hundreds of thousands of elite.

A Selection of Sources:

David Scott Gehring, Anglo-German Relations and the Protestant Cause

Brennan C. Pursell, The Winter King: Frederick V of the Palatinate and the Coming of the Thirty Years’ War

Peter Wilson, The Thirty Years’ War: Europe’s Tragedy

I am not knowledgeable to address the whole thing, but regarding how much free thinking was allowed, you should check out The Sleepwalkers from Arthur Koestler, which is about the Copernicus-Galilei era. TL;DR much more than what you probably think.

You seem to be dedicated to see the whole thing as a sort of a machiavellian power game and not much else. This is actually a very entrenched perspestive today, which is sort of hard to challenge without really throwing a library worth of nuance and "one one hand, on the other hand"-ery on you, let me put it this way, power games obviously happened, a lot, if you think the people on top were generally cynical and did not actually believe in all this, what else do you think they believed? I mean, today atheism largely means taking god out of the gaps, as there are only gaps left in a world well explained by science, but how would that actually work back then? What else to believe instead? http://www.amazon.com/Hobbes-Dictionary-A-P-Martinich/dp/063119262X argues that when Thomas Hobbes was accused of atheism, that largely meant back then "belives in god but not divine providence".

Or for example, seeing heaven and hell as a medieval inventions for power-gaming, actually it would be hard to find any religion ever that would not propose that how morally a person lives has an affect on what happens to him in the afterlife, and already the Osiris cults in Ancient Egypt would paint a familiar scenario: http://en.wikipedia.org/wiki/Hell#Ancient_Egypt

I would argue what is very contradictory about the purely power-gaming model is the implicit assumption that there was a population who actively wanted some kind of an egalitarian society and this was pressed down. The issue is, the active desire of an egalitarian society is what cannot be really demonstrated. It is not a human universal. To large extent it was invented by secularizing various religious ideals, like the dignity of a person or equality in the eyes of god. Do you think an atheist or lightly religious society would never accept inequality or opression? They could simply accept it on an amoralistic basis, that the strong do what they like and the weak suffer what they must. No question that religion, culture, myth holds a society together and helps justifying power relations, but still actual power matters a whole lot more. A good historical example is excommunicated kings. By your logic religion harmed, and not helped in justifiying their power. It often did and sometimes they cared. But very often didn't. It is just enough to fear a ruler, and basically think your life expectations are generally better if you don't rock the boat much, it is not necessary to se him as just and valid. I think you are reading back a brave and dedicated democratic spirit that would be very anachronistic.

The Early Modern Period in Europe was, quite possibly, the era of the cruelest religious persecution and bloodiest religious conflict in history. The Thirty Years War spilled far more blood than the Crusades. Was it all about something else -- something other than religion? What appeared to be religious persecution was really just the persecution of worldly political enemies, and what appeared to be religious warfare was really just the usual warfare that had always plagued Europe? To some extent, probably yes. (After all, in the Thirty Years War, France did side with the mostly-protestant cause.)

Claiming that it was all pious rationalization for other motives, though, doesn't seem parsimonious in explaining everything. Was there some gain for English recusants who stayed loyal to the pope and refused to obey the heretic-queen, even on pain of death? You could probably come up with some sort of rationale, but frankly it'll probably be significantly more tenuous and convoluted than saying that for a significant fraction of the Early Modern European social elite, religion was life-and-death serious business.