The pagan origins thing is a myth, btw—all the various bits which people point to as evidence still date back to a Christian-era Europe and cannot be traced back further. Christianity has been around for a very very long time and has had a lot of time to evolve its own traditions. :)

For example, Christmas trees date back no later than the middle ages:

https://youtube.com/watch?v=m41KXS-LWsY

I think Christmas is whatever one makes of it. Whether or not one believes it's a Christian holiday or Pagan holiday — celebrate/observe (or don't) how you like. My wife's family is Catholic and they believe it's a sacred holiday so they celebrate it religiously. Its origins aren't important to that effect. Many religious traditions can probably be traced to secular origins.

Personally for me, it's a good excuse to take time off work and hang out and feast with people who also have time off work. I personally think that (at least in the States) it's basically a Commercial Holiday, in that it just encourages over-consumption, consumerisation.

Yeah that's probably more common than you might expect. My mom is a teacher at an Islamic school and some of her 1st graders would often say stuff like "we have a Christmas tree at home but my mom told me not to tell anyone," which might give you an idea of how Muslims view Christmas.

It's not something I personally want to do for my family, but I don't think any less of families that do that.