I'm going to directly dive into the second question as the first one has already been answered by u/welfontheshelf. I'm currently away from home and will mostly rely on "Auf der Spüren der Türken" by Georg Schreiber, but it shall suffice.

To start with, the city of Constantinople was already far from being the great metropolitan city that it used to be when the Ottomans have arrived; the Byzantines have lost almost all of their lands and their economy was in ruins, and approximately 50,000 people were left in the city as of the start of the 15th century. This meant that even before the Ottomans arrived, the city was severely underpopulated and required the special attention of sultans to revive the city.

Adding to this was the 3 day long sacking of the city by Ottoman soldiers (which was common for cities that resisted occupation). Although there were a small part of the population which left via the help of Genoese merchants, those were either substantially rich or a part of the Imperial household.

With the relocation of Anatolian Turks to repopulate Constantinople and the city becoming an important trade hub again, the Christian Greeks became a minority. So, in order to answer your question, no most of the Christians did not simply migrate or convert, but there weren't much people in the city already by the time Mehmed II declared it the new capital. They would be eclipsed by newcomers in the next century, although the descendants of those Christian Greeks would continue to be inhabitants of the city until the 1960's.

For your first question, this has been asked several times this week - I'm not sure where this belief comes from, but no, Mehmed did not buy it.

Here is the original question and discussion, which has some books that you might find useful.

Apparently my answer has not been convincing...but I will keep adding to that page, if I can find any new information.