What gets me is that people seem surprised that Christianity has pagan elements in it. Cultures and practices shift from focus to focus. If you know your history this shouldn't be a surprise at all. Early Christianity spent most of there time with the pagans and most of their converts were pagans. However converts even to this day have a problem not separating what they were used to to new practices. Most of the time they would fall back into the old practices. Look how long it took the Jews in the Old Testament to finally give up paganism. The only way you are going to get the pagans to stick around is to bring in elements that they are familiar with but give it a Christian theme without turning Christianity into paganism. Instead of worshiping the snow gods or whathaveyou, worship Christ and celebrate the Incarnation. You simply shift their gaze. There shouldn't be anything wrong with this kind of culture adaptation as long as it fulfills the requirements of Christianity. The idea Christmas is null and void due to its pagan elements is a bit shallow. Judiasm also has pagan elements and so does Islam. Christianity as it went brought into it's teaching elements of the Enlighenment, the Renaissance, and Scholasticism.
When it comes to the Incarnation and birth, they really don't know when Christ was born. They have ideas but no exact date. Again, this shouldn't come as a surprise. It is rare to have exact dates of birth of individuals of that era unless you were a part of a society that kept excellent records, like the Romans did. But Jesus was born amongst the poorest of the poor.
Also the celebration of Christmas was most certainly not there from the beginning. Why on Earth should anyone expect that to be the case. Remember, what we have is an event and the witnesses of the event traveling to different places to spread the word of the event. You have Paul who had a LOT more resources and was able to travel farther than most spreading the Good News. At this time nobody was really reflecting on the early life of Christ, most of the reflection was on the Passion, Death and resurrection, an event that they witnessed. One the biggest reasons for the Gospels to be written was due to the death of the Apostles and therefore scholars were afraid the history would get lost unless it was written down. Thus comes the four gospels we know now and plenty of others that are not a part of the Bible. The Bible was put together due to the numerous gospels and writings and problems of followers not knowing what is "true" and what was made up by some wacko. Thus is makes sense Christmas came a little late, around the 5th century as the Church was a bit busy with heresies, persecutions by Roman Emperors who got a little fed up with the Christians not pulling their weight and getting a definitive text down for all the bishops to pass on to the faithful. Even then this idea that Christmas was just invented is shallow. Reading texts on the Incarnation should help anyone understand the kind of weight Christians put into Christmas.