Both yes and no. In theory, the conquered peoples were allowed to continue being Christians, but they would often be subjects to discrimination, or even attacks by fanatics. There have also been cases when the Turks tried to violently convert people to Islam, mostly during times of civilian uproar (before the successful revolution of 1821, there have been many failed revolutions many of which were started by priests and monks, like Archbishop of Kefalinia Dionysios the Philosopher).
Of course. They had some rights (not quite as many as Muslim converts had but...) and were not considered animals. As much as we hated each other, the Turks (and Mehmed in particular) admired greatly the Byzantine/Greek civilisation. After all, Mehmed was self proclaimed "Kaesari i Rum" meaning Ceaser of the Romans/Greeks.
Not very good. Just to give you an idea, the Greeks of Constantinople used to be over 120.000 in number during 1927 (and that was after the Pontian Greek genocide of World War I). Now, after discriminations and pogroms, they have been reduced to merely 4000.
As a Greek, I've been having the same problem as you, but what is done is done.
I suggest we try to improve our relationships as peoples, not keeping grudges, so that we can forgive and be forgiven.
After all that is the meaning of being a Christian to begin with.