I can explain the population of the modern city of Istanbul, but I don't know a particularly large amount about 15th and 16th century Ottoman Istanbul. Istanbul was a city of only less than a million people as recently as 1950, and now it's a city of 14 million officially, with unofficial estimates being generally higher (sometimes significantly higher, though I could never figure out the exact basis of those higher estimates). In a basic sense, most contemporary Istanbul residents are recent Anatolian migrants (both Turkish and Kurdish) who have only come to Istanbul in the last two generations. People from older families might talk about how many "göbek" (literally "bellies") their families lived in Istanbul, but they are a minority.

In about 1927 (the year of the first modern Turkish census), when the population was about 700,000, non-Turkish and Kurdish minorities, especially Armenians, Greeks, and Jews, made up a more significant portion of the population. That census found 110,000 Greeks; 82,000 Jews; 77,000 Armenians in the country. These didn't all live in Istanbul, but the majority of them did. See Icduygu et al.'s "The Politics of Population in a Nation-Building Process" for the decline of all the groups (today, there are probably 3,000 Greeks, maybe 15,000 Jews, and 70,000 Armenians, almost all in Istanbul). There's also small populations of Latin Catholics (Italians) and Bulgarians, with most of the Latin Catholics ariving before the conquest of Istanbul. For more on minority groups in the Turkish censuses generally, see Fuat Dundar's book which I believe is available from his academia.edu page. I know least about the Greeks, but I would guess that many of them claim descent from original inhabitants of the city or region, before the Ottomans conquered the city in 1453. The Armenian population of Byzantine Constantinople was small, so I assume most of the Armenians' ancestors probably migrated to Istanbul in Ottoman times. There was a small population of Greek-speaking Jews in Constantinople, but a much large population of Spanish-speaking Jews arrived in the 1490's as the Jews were expelled from Spain and Portugal (the confiscation of their property is one of the things that helped finance Columbus's trip).

There was also a steady trickle of minority conversion to Islam, and the bulk of the Turkish population probably has some Armenian, Jewish, or Greek heritage (as well as Arab, Persian, and perhaps Caucasian and Balkan). The most famous Ottoman architect, Mimar Sinan, for example was probably an Armenian convert to Islam. The conversion process was not only a religious conversion but also an ethnic one in the Ottoman one, nationality millet was religiously defined--to be Armenian was to be a specific type of Christian, and to be a Turk was to be Muslim, so switching religions from Christian to Muslim essentially meant turning from an Armenian into an Ottoman/Turk. The marriage boundaries (which are still largely intact today, despite decades of civil marriage--Zeynep Ozgen has a recent article on this) remained important, so a Muslim would assimilate through marriage into the Muslim population. A Muslim would also be allowed to take minority wives, but the offspring of these marriages would be Muslim, and integrate into Muslim society. Many of the mothers of Ottoman sultans, for example, were not Turkish speaking, coming especially from the Balkans, Caucasus, and as far a field as Poland (Roxelana/Hürrem Sultan is generally though to have been Polish, Rusyn, or Ukrainian). The women of the imperial harem came from a variety of countries and "extra women" from the harem were often married off to important young Ottoman officials trained in palace schools and sent off to administer the periphery. Other elite Muslim families from Arabic and Persian-speaking areas would easily intermarry with Turkish-speaking high elites in Istanbul. The exact genetic make up of modern Turks is tricky to find out, but there's a whole Wikipedia article on it, which covers all of modern Turkey, not just Istanbul.

First of all, there were comparatively very few people left. All told there probably weren't any more than 50,000 present for the siege. Those who get out largely did, and the fighting force inside the walls was probably somewhere around 7,000 total.

On the final day of the siege, people who could manage to find their way to the ships did and were able to escape, but far more could not. There were crowds packed into the Hagia Sophia and the Church of St. Theodosia, whose feast day it was (that still exists, by the way, as Gül Camii or the Rose Mosque).

As for what happened to them, during the general pillage that followed the immediate fall of the city, they were captured and those who could be ransomed generally were. For example, the eyewitness chronciler and close companion of Constantine, George Sphrantzes, had to ransom his own wife post-siege. One of his daughters actually went into the sultan's harem.

Others were treated as their perceived station and usefulness directed. The monk Gennadios Scholarios, renowned for being an anti-Unionist, was chosen as Patriarch and accounts of his relationship with Mehmet show that they were friendly and respectful. Granted, I'm sure this had much to do with the fact that anti-Unionism (being opposed to the union of the Roman and Orthodox churches) was seen as a good thing by the Ottomans, who were loath to bring the forces of western Europe to bear on their newly-conquered capital, which might have happened if the two sides had fully considered themselves of the same religion.

As the dust began to settle, there were Greeks left in the city, and a powerful Greek merchant class began to emerge, some of whom even occupied high positions in the Ottoman court. They were called "Phanariotes" after the neighborhood where they lived, the Phanar, or Fener in Turkish. These Phanariotes included names of some of the grand old families of late Byzantine aristocracy. Ypsilantis was a Phanariote family too.

Sadly, someone else will have to pick up the thread later, as I'm pretty much useless post-15th century. I do know that quite a lot of population exchanges have happened, so I wouldn't be too sure that there are unbroken lines of Greek families still in Istanbul. But hopefully this was helpful as from the standpoint of what happened immediately after May 29.

This might not be directly related. Based on most genetic testing, unless there was CLEAR account of massive exodus, massive immigration or genocide, the general rule of thumb is the ancient native population of an area are almost the major genetic contributor of the current population. Including many cases people do to realize to be true. British are more genetically more "Celtic" than Anglo-Saxon. The Turks are more genetically "Native Anatolian" than "Turkish". The Palestinian are more "Hebrews" than Arab. The Egyptians are more "Ancient Egyptian" than "Arab". Most conqueror's goal for killing was to rule not Genocide. Most population do not go on exodus, because the risk. Most invading population are really invading armies, which usually only comprise of very small number of the invading population.