When was Jesus birth and death?
The Bible doesn't record exactly when Jesus was born. One thing is clear, however. The date when we celebrate His birth is almost certainly not the correct date.
That isn't to say that it is wrong to celebrate His birth. His choosing to come and die for us gave us all a means to salvation outside the Jewish law, and we should all celebrate what God did in coming, living as a human, and being sacrificed for our sins. That action sets our Christian God apart from every God mankind ever dreamed up on their own. The start of Christ's life is a good day to celebrate, much as we celebrate the end of His life and His resurrection from the dead during the Easter week.
But the actual date chosen to celebrate the beginning of this event is wrong. I can say that with certainty because Jesus was Jewish. This matters because the Jewish calendar does not match up with our calendar. The Jewish calendar follows the lunar cycles, with each month being 29 or 30 days. It adjusts to the solar cycle by periodically adding a month to get back in sync with the yearly orbit of the Earth around the Sun. Quoting from the public domain Calendar Converter...
The Hebrew (or Jewish) calendar attempts to simultaneously maintain alignment between the months and the seasons and synchronise months with the Moon - it is thus deemed a "luni-solar calendar". In addition, there are constraints on which days of the week on which a year can begin and to shift otherwise required extra days to prior years to keep the length of the year within the prescribed bounds. This isn't easy, and the computations required are correspondingly intricate.
Years are classified as common (normal) or embolismic (leap) years which occur in a 19 year cycle in years 3, 6, 8, 11, 14, 17, and 19. In an embolismic (leap) year, an extra month of 29 days, "Veadar" or "Adar II", is added to the end of the year after the month "Adar", which is designated "Adar I" in such years. Further, years may be deficient, regular, or complete, having respectively 353, 354, or 355 days in a common year and 383, 384, or 385 days in embolismic years. Days are defined as beginning at sunset, and the calendar begins at sunset the night before Monday, October 7, 3761 b.c.e. in the Julian calendar, or Julian day 347995.5. Days are numbered with Sunday as day 1, through Saturday: day 7.
The average length of a month is 29.530594 days, extremely close to the mean synodic month (time from new Moon to next new Moon) of 29.530588 days. Such is the accuracy that more than 13,800 years elapse before a single day discrepancy between the calendar's average reckoning of the start of months and the mean time of the new Moon. Alignment with the solar year is better than the Julian calendar, but inferior to the Gregorian. The average length of a year is 365.2468 days compared to the actual solar tropical year (time from equinox to equinox) of 365.24219 days, so the calendar accumulates one day of error with respect to the solar year every 216 years.
This is a very different algorithm from our Gregorian calendar's pattern of varying length months and a leap day added every four years to February, unless the year is also evenly divisible by 100.
What the different algorithms mean is that any particular birth date marked on the Jewish calendar will move around on our calendar each year, just as any special date on our calendar will move around on the Jewish calendar. That is why Easter, which is tied to the Jewish Passover, is not the same date each year on our calendar but moves. Passover is always constant, however, on the Jewish calendar. The same would be true if we knew the date of Jesus birth, and I think we would all agree that if we were going to celebrate someone's birthday we should do it in the calendar system they use, even if it is inconvenient for us. So to accurately celebrate Christ's birth, we would celebrate a different day each year on our calendar, just like we celebrate the discovery that He was resurrected from the dead on a different day each year.
An individual with much more time than myself did an analysis of celestial events, Jewish feasts, and the details recorded in the Bible about Jesus birth, and came up with a birth date for him of Tishri 15, 3757 and a death date of Nisan 14, 3790. Whether these dates are correct or not will only be known once we get to heaven unless God chooses to reveal the fact to us. These two dates correspond to September 14, BC 4 and April 3 CE 30 on the proleptic Gregorian calendar. Other individuals have done other calculations to come up with two or three other possibilities (one in CE 28 and two in CE 31). There is evidently a letter from Pontius Pilate written on the 5th of the Kalends of April, CE 30 describing the crucifixion which might lend weight to the CE 30 date. Geologists have also dated an earthquake using sediment analysis to CE 31 +/- 5 years, so that would lead weight to the CE 31 date.
You can use the above calendar converter link to see what those Jewish month and day combinations correspond to on our calendar this year. Jesus was crucified around 3 p.m. so the days would match up with the Gregorian equivalent of the day for His death. Regardless of the year, the day is almost certain to be Nisan 14 on the current Jewish calendar, moving with respect to the Gregorian calendar every year. So the concept of Good Friday and Easter Sunday is not historically accurate. It is good to recognize his resurrection and especially his death. It is his death that saves us. It is the stripes he took and the suffering he went through before his death that heals us and gives us strength. The resurrection gives us a hope for the future, but it is a future that is only ours if we accept the saving power of his blood.
We don't know exactly when during the day He was born, so if you wanted to get technical, you would need to know that specifically to pick which of the Gregorian dates to match, and if you wanted to match the exact time (for either birth or death) you'd need to adjust it for your time zone difference from Jerusalem. Regardless, his birth was most likely in the fall of the year when shepherds actually might have been out in the fields in Israel and definitely not at the pagan celebration day for the shortest day of the year which the early church chose to celebrate.