TL;DR info at the bottom.

About 500 years BC, pagans would celebrate the coming of the middle of winter (December the 25th) by feasting of from 17th till the 25th. During this time, they would place a tree in or at their homes, decorating it, having big meals, crafting or looking for gifts for one another, and would celebrate the whole idea of being together, warmth, the light, and that soon, they would be heading towards summer again, which, for pagans, without radiators etc. was a pretty nice thing. Years and years later, Christ would be born, not even on the 25th probably, but he was later on said to be, merging the pagan holiday with the christian one. This probably done because it was such a perfect day for it, Jesus was born on the one day in the year we start heading towards the light and warmth of summer again. Back then, the two weren't mixed yet, just practiced at the same time, but later on, the two customs would become one. Only the star placed on top of the tree was added by Christians, but other then that, decorating trees has, and will always have pagan roots, and nothing to do with the birth of Christ. It is even said that any forms of celebration in the form of holidays and parties, has nothing to do with Christianity, and are all pagan customs. Therefor, some Christians are actually offended by people celebrating under the name of Christ, even tho there is no link between the two.
TL;DR: Pagans had Christmas 500 years before Jesus was born.

GIf unrelated. completely.

the idea of a bearded man giving presents to good behaving people and punishing the bad derives from odin doing just that (only that he rode on sleipnir's back, not in a sleigh)

Google "sun gods", and learn how "Christmas" is only the last name of a very ancient holyday :)