Mehmet II never wanted to destroy the city. He plainly wanted it as his capital. He repeatedly tried to convince Constantine XI to yield (at one point he even resorted to outright bribery, offering Constantine all of Morea if he yielded the city; Constantine didn't bite and would [probably] die when the city fell), so that he could deny his soldiers the traditional sacking that would follow a city captured by siege. He also bent the rules further by pretending that individual walled villages behind the city walls, of which there were plenty, were permitted to negotiate separately. Those that surrendered received garrisons to protect them from Mehmet II's own looting troops.
So, to answer your first question: yes. There absolutely were Romans left in Constantinople when the walls fell.
For the second half of your question, the answer is a bit more complicated and ultimately comes down to the Ottoman tradition of religious tolerance. While there are incidents of mid-level officials acting out against religious minorities (Ottoman Egypt certainly had a mixed reputation after Wahhabism gained influence in the 18th century), the Ottoman Empire in the 15th century was a remarkably tolerant place. As long as you paid your taxes and kept the peace, the Ottomans didn't care terribly much about your specific faith. They even implemented a series of religious courts (millets) to govern legal disputes between people of the same religion.
To that end we also see Mehmet II turn around an appoint a new Patriarch of Constantinople (Grennadius II, who was also the de facto leader of the Orthodox Greek people within the Ottoman Empire) in 1454. Hagia Sophia would remain a mosque until the 20th century (and may become one again), thus underscoring Islam's primacy in the Ottoman Empire, but Christians certainly had their rights.
Ultimately, the Ottoman government was more interested in stability and a functional bureaucracy than it was in matters of religion or ethnicity. While it's true that being Muslim was often required to advance to high rank, it's also worth noting that some of the most powerful officials in the Ottoman Empire weren't even Turkish (and the most famous political dynasty, the Koprulu family, was Albanian). In fact, the five Grand Viziers immediately following the Conquest of Constantinople in 1453 weren't Turkish, though they were Muslim.
Basically, if you weren't unfortunate enough to be swept up by the devsirme system, your religious faith was your concern, not a state concern.
According to a census carried out by Kadi Muhyiddin in 1478, the population of the City at that date included the following elements:
Istanbul (number of families):
and in the Galata district, the numbers are:
The military class, enjoying immunity from tax, was generally left out of such surveys. In any case, it is clear that the Romans who were recorded as Orthodox Christians in this census were the largest minority after the Moslem majority.
Source: The Re-Building of Istanbul By Sultan Mehmed The Conqueror, Halil Inalcik, 1967.
The article which can be dowloaded here is strongly recommended for reading.
In fact, Mehmed II had ordered another survey in 1455, just two and a half years after the conquest. Halil Inalcik's paper, which can be dowloaded here, gives more details of how the inhabitants of the City were treated after the conquest. The City was taken by storming, and so it was looted. However, the Galata district was taken by a covenant and spared from looting.
There was, under the Ottomans, a large Greek population in what is now Turkey, in addition to Greece itself falling under Ottoman control. Many of these Greeks were forced to migrate to Greece as a result of the war between the new Greece and the new Turkish state after WWI, and the terms of the treaty that ended it. Before that time, though, there were plenty of Greek Christians in the Ottoman empire, including in Istanbul itself, so presumably the answer is that the Romans who remained at the end of the siege (and survived the following takeover and looting) formed the core of this population.
Additionally, the Ecumenical Patriarch of Constantinople continues to this day to be the nominal head of the Orthodox church-- though his actual authority and role in representing orthodox churches outside Turkey and Greece is a bit murky. This post has resided in Constantinople essentially continuously since ca. CE 450, right through the siege of Constantinople.