Just so that you know, I agree with the stance you take on the Christmas tree. The Jeremiah passage can be used for trees because all the steps are there except the part where they fashion it into an idol. But I always quip that if we decide to skip a step, and just keep the tree intact, it makes it any better? LOL. The principle still applies because the Lord warns us not to learn the ways of the heathen, in which the tree is part (not just idols). There is adequate warning throughout scripture not to mix worship, even if done in sincerity.
The way I handle this holiday with the rest of family who wants to celebrate it is to just consider it a secular holiday with no personal worship but only have the decorations and gifts, akin to having parties and shooting off fireworks on the 4th of July. I go along with them because that's them, but I don't initiate it. In other words, if I lived as a hermit or single, I would not really bother beyond making the others happy but nothing for myself. I think it's a hokey sort of holiday.
As for "picking a date" to celebrate when Christ was born, might as well use the bible since it indicates it was in September, the feast of Trumpets, not a worldly date used in ancient religions. But really, God does not want us to celebrate birthdays, but rather use these same feast days that reveal his redemptive plan of salvation to mankind, be they now in this age, the millennium or the age to come.
Most of the tree, garland, branches, Santa's and whatnot seem to have a Celtic base to much of these, the same as Halloween. They are sibling holidays -- so celebrate one, celebrate both (which means I'm giving kids who want candy for Halloween some latitude when they are warned it's the "devil's holiday"). As a matter of fact, some say the most haunted night of the year is not Halloween night, midnight on November 1, but Christmas eve, December 25 (or more accurately, the winter solstice), thus the "scary ghost stories" that Andy Williams sings about and old Victorian themes like the "Christmas Carol". Halloween seems to just be the initiation of bigger spiritual activity later.