What was the fate of the Byzantine people after 1453? Were they all killed, sold into slavery, or able to live in peace?
What does it mean to be 'Byzantine', when & how did that term come to be? How did the Eastern Roman Empire become to be known as the Byzantine Empire?
How/when did the Eastern Roman Empire switch to being a Greek dominant oriented language instead of using Latin?
Why did the Byzantine Empire switch to the Eastern Orthodox Church instead of Roman Catholic Church?
I can offer a general perspective which I think will answer your question.
Greek was always the dominant language of the Roman east. Justinian (d. 565) was the last advocate of Latin, but Greek soon became the state language.
Byzantium never "switched" to anything; it maintained stricter adherence to the original church councils (Nicea, Chalcedon, etc.) as opposed to the Catholic west which continued to innovate. The east and west had long drifted apart when in 1054 the pope excommunicated the Patriarch of Constantinople for supporting an interpretation of the Trinity that he did not agree with. The Orthodox claim that they never deviated from the norm, and it was the west that did. They have a real argument.
What happened to the Byzantine people after 1453? I would ask who were "Byzantines" in 1453? Greece (the heartland) had been conquered by the Ottomans 50 years ago, and there were very few "Byzantine" areas left. Trebizond in the east continued for a few decades; Serbia and the Balkan nations tried (in vain) to maintain independence. But as to the Byzantines that remained in Constantinople, it was a mixed fate. Some went west and played a role in the foundation of western humanism; some stayed in the east and adjusted to Ottoman rule.
It's a hard question to answer and I hope my general points help.