I just want to note that the use of the term Islam such as in this question is not accurate. Islam is a religion, not a state.

Can you give a particular time you're interested in?

After the Arabs conquered Egypt, they left the Orthodox Patriarchs in place, for the most part. Indeed, the Eastern Patriarchs were in frequent communication with their colleagues back in Constantinople and even participated in church conferences from time to time.

Early Islam did not actively seek to convert conquered peoples. In fact, the Arabs went out of their way to not even mingle with the Christians. They would build their own settlements near, but not in, the existing cities and pretty much let the Christian societies do their own thing--so long as they paid their taxes.

To drastically over-simplify a relatively complicated religion, Islam views itself as the successor to both Christianity and Judaism. If these religions were Pokemon, Judaism would be the initial form and Islam the final form (I can't believe I just used that analogy in serious discussion). The teachings of the Quran further recognize various prophets in the other two religions as prophets of Islam. Jesus, for instance, is venerated in Islam (though he is not recognized as the son of God).

As a result of this outlook, Islam considers Christians and Jews to be "People of the Book," practicing a vaguely similar, if less accurate, form of Islam. And because these People of the Book aren't pagans (like the initial Arab tribes conquered by Mohammad), Islam affords them a variety of rights and benefits. To over-simplify the vast array of Islamic laws usually clumped together under "Sharia Law" (as if a religion of a billion-plus adherents could ever have a single school of legal thought), if the People of the Book paid a special tax (a jizya), and if they kept the peace, they were entitled to protection and respect from the Muslim authorities. The People of the Book weren't entitled to the same sort of benefits as Muslims (the early Muslim states had one of the first forms of social welfare but, as it was largely paid for by the zakat, it was only available to Muslims), but they were permitted to worship in their own fashion and carry on with their lives as long as they kept the peace.

In fact, the Quran explicitly forbids forced conversion: 2:256, "There shall be no compulsion in [acceptance of] the religion. The right course has become clear from the wrong. So whoever disbelieves in Taghut and believes in Allah has grasped the most trustworthy handhold with no break in it. And Allah is Hearing and Knowing."

That's not to say it never happened. The Ottoman devsirme system absolutely did involve indoctrination and conversion of Christian boys so that they might be considered perfectly loyal to the state (these boys would go on to become the infamous Janissaries). That's not the only such example, but cases of forced conversion such as that were exceptionally rare when you consider how many non-Muslims (dhimmi) were part of the various Islamic empires.

While I can't really speak to how the various Islamic caliphates treated the three sees you mentioned, I can speak to how the Ottomans treated them. Mehmet II conquered Constantinople in 1453. The following year he elevated a new patriarch (Gennadius II). From that point on, the Patriarch of Constantinople was millet-bashi ("millet" refers to legal courts that applied religious law to cases where all parties involved were of the same religion; millet-bashi thus is the spiritual leader the people represented by that millet) of the Orthodox people within the Ottoman province of Rumelia, which roughly corresponds to modern Greece and European Turkey, regardless of whether or not those people were ethnically Greek. The Ottomans never returned Hagia Sophia, though. That's still a mosque.

Similar policies were enacted in other Ottoman provinces to correspond to other religious groups. The Syriac Orthodox Church, Nestorian Church, Armenian Orthodox Church, Armenian Coptic Church, and so forth were all given their own legal protections and, later, their own millets. The Bulgarian Orthodox Church got its own millet in the late 19th century and at some point there were also Protestant millets.

As for the Patriarchs themselves, they were treated reasonably well by the Ottoman authorities. While they were not Muslim and thus never quite equal in social standing to imams or the like, they were usually well-respected by their respective communities and showing them respect meant that they were less likely to rabble-rouse (and thus less likely to foster instability).

But the policies of the central government were often shaped by the mid-level officials in the area. A great example of this would be the Coptic Christians of Egypt. In 1748, Patriarch Murqus VII led a pilgrimage to Bethlehem from Harat al-Rum (a Christian quarter in Cairo). This incensed a number of locals, who felt that the Copts were emulating the Muslim hajj. Eventually, and apparently before they even left Cairo, the offended locals managed to get the local Janissary commander and the chief of police to prevent the pilgrimage from continuing -- they attempted to head off the pilgrims and seize their goods, thus forcing them to give up the "Christian hajj," but the Copts were apparently tipped off and eluded the crackdown. The main complaint was that Ottoman law forbade public displays of this sort by religious minorities.

The Copts tried again in 1751 but this time they sought legal support first. They consulted with Sheikh al-Shubrawi, who issued a fatwa declaring that they were not to be hindered in anyway. This went well for about an hour and then the sheikh was forced to defend his decision in public. He was accused of being bought and paid for and apparently gave in, permitting the gathered crowd to loot the caravans and chase the Copts off with sticks and stones.

It's worth noting that these events, as detailed in Coptic Christianity in Ottoman Egypt, occurred at around the same time as the emergence of the fairly puritanical Wahhabist school of thought that is the dominant school of thought in modern Saudi Arabia. This also comes shortly after the conclusion of another Ottoman-Persian war in the 1740's, which certainly didn't help matters.

This is a ridiculously broad subject and I could keep talking but I think you get the general idea by now. The central governments usually favored tolerance and more-or-less-equal rights while local governments may be pressured by their citizens into being less tolerant of religious minorities, especially if those minorities were engaging in public displays of their religion.