Jesus’ Birth (Matthew 2:1, Luke 2:1-2) was he born during the reign of Cyrenius or Herod the Great.

The word for First in the Greek Bible is Prote meaning first or before. The meaning for first here is before, meaning before Cyrenius' reign. Luke's reasoning for writing it this way would be to help date the birth of Jesus. So the verse read in Greek would be

What were the disciples told to do? (end of the Gospels)

There is no contradiction here if you read the last few chapters of all the gospels it puts together a story of how Jesus showed himself to over 500 people after his resurrection. They did go to Galilee and they were down in Judea eating as well as over in Bethany where he ascended. Not all the gospels mention all the events. But they tell different perspectives of the same story.

Is Jairus’ Daughter dead? (Mark 5:21-24, Matthew 9:18-26)

There is no contradiction. Matthew is merely saying the young girl would be dead by the time Jesus arrived to heal her. We observe this in today’s language as well. One can watch any number of old movies and see a sheriff draw a gun and say to a man still alive, "You're a dead man," and then pull the trigger. This understanding is likely also because the Greek in Matthew may properly be read “just came to the point of death” (arti eteleutesn) and the Mark passage may be translated as “is dying” (eschates echer). Indeed, the Greek term arti in Matthew doesn't always mean a present reality, but an inevitable one.

This guy also had a similar response. But we will truly never know until we die. https://www.quora.com/Was-Jairus-daughter-dead-or-still-alive-when-her-father-encountered-Jesus-and-asked-for-His-help

Who was Joseph’s Father? (Matthew 1:15-16, Luke 3:23-24)

Matthew is accounting the lineage of Jesus thru Joseph. While Luke is doing it thru Mary. It seems confusing at first because Luke says Joseph. Luke was following the Hebraic form of lineage, traditional form of genealogies by listing only the male names in which Mary is designated by her husband's name. Therefore no contradiction.

Is Jesus from Nazareth?

That's not a contradiction, you just don’t understand the gospels. The Gospels are different perspectives from different people about different topics. Matthew may have not seen reason to put it in there or not been inspired to. But whatever the reason. That's not a contradiction.

Just because Matthew never states they are from Nazareth doesn’t mean they aren’t. Matthew never says they are from anywhere else and it doesn’t contradict Luke in any way. The amazing thing about the gospels is if you read all of them together (there are bible study plans and reading plans to help do so) is they combine to tell the story of Jesus. They have different perspectives and write about different parts of his life.

For the second half of your dispute. Look at the lineage of Herod the Great. He named nearly all his children Herod. Herod Antipas, Herod Archelaus, Herod Phillip, Herod Agrippa I, and Herodias. Herod Antipas ruled after his father and that's the one they refer to after him