For the trekking in winter - many of the high mountains passes are obviously closed. But there are specific winter treks. See here for example:
http://www.alpine-bhutan.com/Samtengang_Winter_Trek.pdf
Then altitude sickness is no big deal. You are much lower than the typical trip in Tibet; and you will take it easy. Remember you are hiking up the mountain, not just flying in. No big deal if you take it easy.
Plus - and that's now the advantage of doing a trip to Bhutan. As you have to spend anyway money with a minimum - the trekking trips come with guide/porter. Quite nice - and if you pack right, you just have a light camera bag, a bottle with water, and your CF cards
. Sounds easy.
If you are not fixated on December - September/October is the better season. And truly enjoyable.
Plus don't just do hiking. Make sure you hit one of the towns when a festival is on. There are tons of them - plan your trip around it. Your shutter/lens will thank you.