December 25th (with Christmas Eve on the 24th) was chosen as the birth date of Jesus to correspond with the pagan celebration of the winter solstice. It appears that it is easier to establish a new religion when people respond to holidays that are similar to the ones they already know. The date was established in 274 AD.
Trees in general have been considered as symbols of life, going way back to the Egyptians. Since hanging branches and decorating them during the winter festival called Saturnalia was already a custom, Christianity only followed what was alreeady known. The tree was hung upside down from the ceiling during the middle ages, probably consistent with the former custom of hanging branches. Only later was the Christmas tree displayed in the way that we know today, and the current system caught on quickly.
Mistletoe also has a very long tradition. Since it was recognised for many healing attributes and associated with fertility in old mythology, mistletoe simply became incorporated into Christian practice. Kissing under the hanging mistletoe went from maintaining goodwill, to stealing kisses from women who stood under the hanging mistletoe (and they couldn´t refuse, since that would bring bad luck), to promises of marriage.
The custom of giving and receiving gifts during Christmas also reaches back to the ancient Saturnalia celebrations. Giving gifts is also mentioned in the Bible - the three wise men (The Magi) gave baby Jesus precious presents. Nevertheless, the Bible says nothing about giving gifts to each other. As is true for many other customs, here, too, ancient pagan practices were incomporated into Christianity. Beginning with small gifts presented only to children, the custom of giving gifts spread among adults, too . . . leading straight to the commercialized version of Christmas that we know today J.
The first Christmas card was introduced in 1843 in England, in US the custom dates back to 1850. Cards became soon very popular. They also played a role during the wars, when funny depictions were popular to cheer up the soldiers.
Boxing Day has existed since the Middle Ages and achieved renewed popularity in the 19th century, when the aristocracy gave gifts or extra money in boxes to their servants as a form of appreciation. Nowadays, the Boxing day is largely a commercial holiday of large sales.