I'll answer your second question; the short answer is probably neither, but we don't really know.

Here's the longer answer: The guy who picked the date for 1AD (when he thought Jesus was born) was a 6th century monk named Dionysius Exiguus. But we actually don't know how he chose the date, because the records we have just have him talking about Easter and his year system, not why he thought Jesus was born then.

The most likely method, however, is based on the Gospel of Luke. According to Luke, the word of God came to John (the Baptist) in the wilderness in the 15th year of Emperor Tiberius. This is 29AD (or possibly 27 AD, as he reigned as co-Emperor with Augustus for 2 years before becoming sole Emperor, but that's not how Emperor's reigns were normally dated). Jesus appears in the Gospels to be baptised by John not a huge time afterwards, though no specific dates are given (a few months? A year? Two years? 5? It's not stated). Luke also says that Jesus was about 30 when he started preaching. So, if Luke is right that John started preaching in 29AD, and right that Jesus started preaching when he was 30, and this was 2 years after John started preaching, then Jesus would have been born in 1AD. So perhaps this is where Exiguus got the year.

But Luke also claims that Jesus was born during the census of Quirrinus (governor of Syria), which other sources say was in 6AD, when the area came under direct Roman rule and was integrated into the province of Syria.

And then we have Matthew, who gives a slightly different story about Jesus's birth. Luke claims Jesus's family was from Nazareth but had to go to Bethlehem for Quirinnus's census (which is odd, as that's not generally how Roman censuses worked). But Matthew claims that Jesus was born in Bethlehem during the reign of King Herod. He says King Herod tried to kill Jesus and family, so they fled to Egypt. Eventually King Herod died, but his son Archelaus ruled in the Bethlehem area, so they had to go somewhere else; a town called Nazareth. The only Herod with a son called Archelaus was Herod the Great, who died in 4BC. Thus, Jesus must, if we go with Matthew, have been born before 4BC.

But, as you can see, the two sources don't agree. Their dates are within 10 years of 1AD, but clearly someone's made a mistake (not really surprising considering that both works were written many decades after the events, and this in a period where many ordinary people wouldn't have been able to give their exact date of birth anyway). And of course, there's the fact that both accounts may be largely fabricated to justify how a man from Nazareth could be the Messiah that many Jews thought would be from Bethlehem. So Jesus escaping from Herod the Great (showing he was born before 4BC) or Joseph and Mary going to Bethlehem for the 6AD census, may be just made up.

All we can really say is that Jesus was probably born* around the turn of AD/BC, but we can't really be specific.

*This all assumes Jesus was a real person. Most scholars of the subject say he was, but it would be remiss of me not to mention that there are some scholars who think he's mythical.