1. The New Testament church was like a Jewish denomination. They didn't think of themselves as a new religion. They were the Jews that had the proper understanding of Scripture. They were the Jews who understood who the Messiah truly was. They were the true heirs to the faith of Israel.
So the denominational HQ was in Jerusalem, and the General Superintendent was James, the brother of Jesus. There were mega-churches in Antioch, Corinth, Ephesus, and Rome. Well, let's call them multi-venue churches, since few individual house churches would have held much more than 50. In fact, you could also think of them as collections of small churches by our standards.
2. I wonder if the house of Mary, mother of John Mark in Acts 12, was a house church. I wonder if it was the house church in Jerusalem where Peter primarily worshiped, since that's where he goes when God frees him from prison. Is this the house where they had the Last Supper, the so called "cenacle"?
Was this also the house where the 120 were gathered on the Day of Pentecost? Then John Mark's family would have to be rich. They do have a servant, Rhoda. If this house was connected in some way to the beloved disciple, a connection to the high priest would befit an aristocratic, priestly family. There is a tradition from the 500s that this was the "mother of all churches" (Theodosius).
James apparently did not live in this house, for he is not there when Peter escapes. But that does not mean that he did not worship there. There were other churches in Jerusalem. For example, the Greek-speaking Christian Jews presumably met somewhere else so that they could worship in Greek.
These believers would ideally have seen whole synagogues believe in Jesus. A synagogue was a gathering at this time, not necessarily a free standing building (thus we can never assume when we read of a synagogue in the NT that we are reading about a building. The synagogue in Capernaum or Nazareth at this time, for example, may simply have been a gathering at a particular public spot, maybe even out in the open somewhere). We definitely cannot assume that the buildings where they take you on tour in Israel are in any way like what was there in Jesus' day.
I imagine that there were some places where entire synagogues believed. In most instances, however, Christian Jews became a para-synagogue movement. They would worship in the synagogue then also gather at other times to eat together (how they did communion) or to pray (e.g., some at dawn, especially on the "Lord's Day," Sunday, to remember the resurrection).
3. I doubt there were any Gentiles in this church. It was an Aramaic-speaking Jewish church. They did not have a rule book on membership but there were assumed membership commitments, namely, keeping the Jewish Law. Let's just say they never had pork for communion. Every male had to be circumcised. The women no doubt did not come to worship when they were menstruating because they would be unclean according to Leviticus.
Someone who did not keep these "membership commitments" would not have been welcome.
They would not have been welcoming to Gentiles in this church. Would Greek-speaking Jews have felt welcomed? I'd like to think they could have attended but I bet they would not have felt particularly warm and fuzzy there.
A group that probably was welcome were ultra-conservative believers, Pharisees who believed Jesus was the Messiah but who still believed in ultra-strict Law-observance (see Acts 15:5).
4. We can understand why John Mark might have had some initial tension with Paul. Mark came from a somewhat conservative background, compared to where Paul was leading the mission. We can also imagine that he spread bad rumors about Paul among the leadership after he left the first missionary journey. We can understand James having some caution about Paul since he had only heard one side for a year or more before Paul was actually able to give his side.
Mark would, of course, come around.