1. Epiphany, or Theophany as it is called in the eastern church, is the 12th day of Christmas. It celebrates Jesus being recognized as the king of kings to the gentiles. There is a lot of information on the wikipedia page.
2. Why do we do presents then? Um... this is more of a broad answer. First I would say that we don't do the Santa thing with our kids. If that made you groan, you might want to quit reading here. We do talk about St. Nicolas the man, and we say that some people believe in a modern day Santa, but it is just a fairy tale.
I don't want Christmas to be about presents. I am appreciative of the gesture, it is a nice display of affection, but I think we, as a society, go overboard. I think it teaches kids to be materialistic, adds to environmental degradation (what with the TONS of plastic, paper and batteries bought, also shipping fuel), and supports manufacturing of American goods in poorer countries. For all of these reasons (I could probably think of more if I took a minute) I don't want to associate that with Jesus's birth.
On Christmas we spend time with family, eat, go to church, just have a nice day off from work. We talk about Jesus and the Christmas story. Christmas means Christ Mass. A special mass to honor Christ, a holiday. We want that to be what it is about and that is it.
So then for the 12 days of Christmas we do something special each day. Either make cookies, or play a game, sing carols... ect. Christmasy stuff. It is fun and stretches that special time out. Honestly though, we get caught up in regular life and often have been like "Oh man, we should X, there are only 2 days left!" It is meant to be a fun and special time, but not to strict.