Was the Old Testament written by the Jews before the birth of Christ, then taken by the Christians and incorporated into their Gospel? And does the Old Testament differ between Jews and Christians?

The Old Testament existed before Christ -peace be upon him- and includes the Torah, historical and prophetic books, poetry, wisdom, and apocrypha. Christians refer to its first four sections as the Old Testament, and the origin of this naming goes back to Paul.

As for the New Testament, it is a collection of the four Gospels and their annexed epistles. It is attributed to eight editors and was named the New Testament in 225 AD by Tertullian.