Not sure about Ethiopia, but all other areas you mention were part of the Eastern Roman Empire at that time. Not disputing your claim though, it was a multi-cultural movement.

(edit: strictly speaking, the Eastern Roman Empire didn't exist yet by that name. It was the eastern part of the Roman Empire)

Christianity spread fast among the poorer subjects of the empire, and frequent persecutions occurred, until 311 CE (maybe slightly off the year), where the edict of Milano by Constantine officially recognised it. When Theodosius issued the edict of Thessalonico in 380, the nicene trinitarian Christianity became the only state religion. About into the 6th century, tensions begun among the eastern post-Roman states and the church of Constantinople, which led to the great schism that happened early into the second millennium.

Whilst the anatolians had sort of a partial continuity into Islam by the middle of the millennium, a process that completed with 19th-20th century genocides of ottoman christians. (Off-topic: I'm reading into comparative genetic reseach about anatolian/near-east peoples, of which I'm a member, which suggests that there is a continuous people since the paleolithic here, that is divided differently at different times in the history because of culture replacements of imperial origins, i.e. Parthian, Armenian, Greek, Roman, Muslim and Turkish empires. But especially the Turkish-speaking, the Armenian-speaking and the Kurdish-speaking peoples seem to be genetically very close to suggest common origin, whereas Anatolian turkish-speakers are genetically closer to brits than central-asians.)

Christianity was swiftly Hellenised under Paul the Apostle, making it at least partially a child of clasical antiquity even at that early stage before it spread so widely.

I guess I don't know what you mean by Hellenised and 'child of classical antiquity'. I either wholeheartedly agree or think there's a categorical error (maybe surrounding the definition of Christianity) in your thought.

If I understand it right, Christianity spread fastest among Greek-speaking populations, rather than the Jewish population. From what I understand, the various letters of Paul and the gospels were all written in Greek, rather than Hebrew.

Those facts are true, yes. I wouldn't characterize them as "Hellenisation" and "child of classical antiquity" is all. Koine Greek was the lingua franca at the time. It makes sense that the writing would be more accessible to the Greeks as well.

Why wasn't Christianity more popular among the Jewish people? The Bible has some theological answers to that question. I wouldn't consider a Greek heritage to Christianity to be a main cause of that though. Jesus and the apostles were all Jews, after all.

True, but that wasn't what I meant when I said that the "Greek spawned Christianity". In my mind, the Greek heritage came later, through St Augustine and other philosphers. They relied heavily on Greek philosophy to construct and validate the Christian view of God and the world.

I wonder though, where christianity would be without Constantine.

Probably fine. Christian theology is mostly pacifist and politically neutral. Some argue that a Constantine-like figure was inevitable, given the growth and oppression of early Christians. If you only oppress people and they don't fight back, you lose legitimacy quickly.