I think Blue Horizon covered most of the questions raised. I will just cover some of the rest
I would think that the split from Judaism happened earlier then the end of the first century. As Blue Horizon mentioned, Christians got kicked out of synagogues and that was recorded in Acts, so it was quite early. But there would have been some synagogues which accepted the message and became Christians.
Nero was the first emperor to distinguish Christians as a religion distinct from Judaism when he blamed Christians but not Jews for starting the Great Fire of Rome 64AD, launching the first persecution of Christians in the Roman Empire.
There has always been a hierarchy and liturgical worship from the very beginning. Jesus had the Last Supper and Acts mentioned its commemoration in homes - that would have been the first masses.
Hierarchy in the Catholic Church consist of the triple orders of bishops, priests and deacons. The first bishops were the Apostles - so, that’s in the Gospels. The appointment of the first seven deacons were mentioned in Acts - one of them, Stephen, became the first martyr. The order of priests split from bishops a bit later - before that bishops covered both functions. By the beginning of the second century, the triple orders of bishops, priests and deacons were pretty much established in most places.
The story was that the Apostles drew lots and it fell to Thomas to go to India. After some initial reluctance (great story that one) he baptised many people there and started one of the oldest Christian communities in the world existing today. I wouldn’t say they were Jewish-Christians as by then the Apostles were clear that they were not really Jewish but Christians. The descendants of those Christians are known as St Thomas Christians or Malabarese Christians (after the coast in India where they live). Due to the high regard for St Thomas, almost every other person their Church seem to be called Thomas or Thomasina. Not unusual for there to have a Fr Tom Thomas to marry Mr Tommy Thomas and Ms Thomasina Thomas in the parish church of St Thomas. :D:D Most of them today are Catholics but not Roman Rite. They are Syro-Malabarese Catholics and Syro-Malakarese Catholics and are as authentic Catholics as we are since they are just as faithful to the Pope.
The Nazarenes were first mentioned in Acts as the label Jews gave to Christians. Subsequently many other peoples (e.g. Muslims) called Christians as Nazarenes (or Nasrani or variation on that name) but the Christians never used it on themselves.
There was a Nazarene sect that emerged later in the fourth century as a semi-Jewish-Christian sect who sees themselves as Jews who accept Jesus as Messiah Son of God born of a virgin. They follow the OT Torah and their own Gospel: they do not use any of our Gospels. They are since extinct but they lives on in mythologies of Dan Brown (of Da Vinci fame) and many Muslim scholars. There are a few Christian Churches today that carry the name Nazarene but they are not related to this extinct sect.