#3 yeshua's real date of birth can easily be calculated if you understand certain things regarding John the Baptist's birth and his father's service in the temple.
St. John Chrysostom does just that in one of his homilies for the Feast of the Nativity.
He assumes that St. Zacharias was the High Priest. St. Luke says he was alone when the angel appeared to him. The only time of the year when the High Priest would have been alone in the Temple with all the people waiting outside was Yom Kippur, i.e. the tenth day the month of Tishri (September/October). Yom Kippur could have fallen some time around the 25th of September that year. (The High Priest would have had to purify himself 7 days before the feast, so the conception of St. John might have already occurred.) When St. Elizabeth was six months pregnant (~25th of March of the following year), Gabriel appears to the Theotokos. Three months later (~June 24th) St. John is born. Add another six months and you have the Nativity of Our Lord.
According to tradition, St. Zacharias was indeed High Priest - he is said to have received the Theotokos when St. Joachim and Anna brought her to the Temple. After the decline of the Hasmoneans, under Roman rule, Flavius Josephus tells us that the office of High Priest lost its political significance and it was frequently assigned to one of the priests by casting lots. Our Synaxaria identify St. Zacharias with the son of Barachia, the High Priest that Our Lord says was slain between the Altar and the Temple (Matthew 23:35). He is said to have been killed because he refused to tell Herod's soldiers where his son was. (St. John was six months younger than Christ, so he would have been martyred with the other Innocents, had he not been hidden in the wilderness, where he grew up.)
St. Augustine, in a beautiful sermon for the Nativity of St. John the Baptist, shows how fittingly it is celebrated at the summer solstice and how from then on the days begin to decrease until the Nativity of Our Lord, according to the very words of the Baptist: "He must increase, but I must decrease" (John 3:30). So, even astronomically the celebration of these feasts was very well thought out: the Annunciation (25th March), the Nativity of St. John the Baptist (24th June) and the Nativity of Our Lord (25th December). The Old Covenant decreases and gives way to the New. The two cousins are close, but there is also a mysterious distance between them (6 months ~ winter and summer). St. Elizabeth is old and barren, the Theotokos is young and a virgin. And so on.
To sum up, the calendar is symbolically perfect. No one could have planned it any better.