I'm going to start with the second question, because it leads into the answer to the first question.

Two relevant pieces of background: 1) Judaism at the time of Jesus was what we call "Second Temple Judaism" which was focused on the temple at Jerusalem, had priests rather than rabbis, and had several distinct factions. 2) While the Middle East was controlled by Romans at this point, it had been controlled by Greeks for hundreds of years before the Romans came. There were very few Latin Romans in the Middle East, but tons of Greeks and culturally Greek Hellenized locals.

Judaism, both in its modern form and in the form that existed in the Roman Empire ranged from an ethno-religion to a somewhat permeable ethno-religion (There were times when Judaism was less like this, but not during the time in question). The significance of this was basically 1) Jews didn't go out looking for converts very much 2) Where Jews accepted converts, the barrier to entry was pretty high, for example relatively few adults are interested in undergoing circumcision, particularly outside of a modern medical context, and Judaism also has pretty stringent dietary and other rules that outsiders often don't want to adopt.

During the time of Jesus there was a great deal of religious upheaval in the Roman Middle East. There were various competing Jewish sects, Greek mystery religions, and other cults. One of the consequences of the religious upheaval and the barriers to entry of Judaism was the rise of a population of Gentiles (non-Jews) who either fully or partially accepted the truth of Judaism and worshipped the Jewish god, but did not take the final steps to become Jewish. This population was called the "God-Fearers".

Jesus was Jewish, and believed himself to be fulfilling the promise of the Old Testament, and his disciples believed that about him as well. The Bible recounts a story when after the Resurrection of Jesus, his disciples were empowered by the Holy Spirit to preach about Jesus to people who were not Jewish who spoke various different languages. Whether or not you believe this story, it's clear that at some point relatively soon after Jesus was no longer on the scene, the Jesus Movement was preaching to both Jews and Gentiles. St. Paul was considered to be the "Apostle to the Gentiles" and focused on preaching to Gentiles throughout the Eastern Roman Empire.

There was a conflict within the Jesus Movement (which was at this point transitioning into the Church) between people who felt that Gentiles who joined the Jesus movement needed to become Jewish and follow the laws of the Old Testament (like kosher and circumcision), and the group that felt that Gentile Christians should not need to become Jewish. The former group was lead by James the Just (Brother of Jesus and proto-bishop of Jerusalem) and the latter was lead by St. Paul. Ultimately St. Paul prevailed approx 30 years after the time of Jesus. This was one of two decisive events that caused Christianity to be distinct from Judaism. The Second was the Jewish War.

The Jews rebelled against the Romans in 66 AD and after a brutal 7 year war they were completely crushed, the Second Temple was destroyed, Jerusalem was basically destroyed and replaced with a Roman city that Jews were forbidden from. Second Temple Judaism as well as most of the pre-existing Jewish factions ceased to exist, and modern "Rabbinical" Judaism, meaning focused on learned teachers rather than a hereditary priesthood, rose from the ashes. Because modern Judaism and Christianity both emerged in the latter half of the 1st century out of Second Temple Judaism, it might be more appropriate to consider them sister religions rather than one growing from the other.

The end of Second Temple Judaism and the decision of Christianity to do away with Jewish barriers to entry led the God Fearers to largely convert to Christianity, which led the demographic transition of Christianity away from being majority ethnically Jewish.

Christianity began to expand across the empire, most successfully in the Middle East, Asia Minor, and Greece, but with significant nodes in Rome, Alexandria (Egypt), Carthage (Northern Africa), and Lyons (France). Christians flew under the radar to some extent for a while, but they were at risk because of their refusal to sacrifice to the Roman gods. Relatively few people at this point were fervent believers in the Roman gods, but refusing to honor the gods was seen as an offense against patriotism, and an evolving philosophy among sophisticated Romans was that there was one god, but everyone was obligated to worship their local traditional gods and the Roman gods in the traditional fashion, and that would get back to the one god. The Jews had a longstanding exemption to the requirement to sacrifice, but Christians being increasingly distinct, did not benefit from this exemption. 1/2