I am, as a Christian, always somewhat amused by the idea that Christians "stole" Christmas. To do that, paganism would have to be a "thing" rather than a collective noun for anything "not us" (like the Jewish "gentile"). Paganism represents whole streams of, at times quite antithetical, traditions, and they freely "stole" from one another (or, at least, were moved by the same impulses to re-invent similar traditions). Thus, Christianity no more stole Christmas than the druids stole their solstice celebration from earlier religions.

No, Jesus was not born on December 25th, but then the queen wasn't born on her official birthday either. The early church wanted a day to celebrate his coming, and people--well people in northern, temperate zones anyway-- seem to need a mid-winter feast to keep themselves going(as blandiver states above), so it just made sense.

As for the symbolism, again all religions borrow what they find useful from their past and bring it with them, and why not?