The Gospel of Matthew says:
When Joseph awoke, he did as the angel of the Lord had commanded him and took his wife into his home. He had no relations with her until she bore a son, and he named him Jesus. When Jesus was born in Bethlehem of Judea, in the days of King Herod, behold, magi from the east arrived in Jerusalem, saying, “Where is the newborn king of the Jews? We saw his star at its rising and have come to do him homage."
Clearly Jesus was already born by the time the Magi arrived in Bethlehem. How much time had passed between his birth and their visit is not recorded in the Gospel.
As for the Shepherds being outside and the date of December 25th, there’s no real way to definitively solve the issue. December 25th was the date chosen for to celebrate the birth of Christ, it is not a historical claim to his actual birthday.