i have travelled on the transsib from moscow to vladivostok (and after that onward to harbin) in january and absolutely loved it. this was more then 10 years ago however, so my info is not really up to date. on the other hand, i dont think that a lot has changed since.
obviously it does get pretty cold (i remember getting of the train in irkutsk, with 35 celsius minus, and a local commenting "you are lucky, today its quite warm"). the train itself is heated and very warm, the temperature difference between inside the train and outside is huge.
as i was not on the trans mongolia route, i didnt stop in ulan bator. also, i didnt stop in novosibirsk and yekaterinburg unfortunately. i did stop in ulan ude, which i very much enjoyed. i stayed with a local family there, visited a buddhist temple near the city, an opera show, and was probably the only foreigner in the city. i also stopped in irkutsk, and that is of course a must stop on the transsib. i would spend 1-2 day in the city, and 1-2 day in a village on lake baikal (listvjanka is the most "famous" one). this is how you imagine siberia to be in winter. the lake is frozen, you can walk on it, even trucks can drive on it. yes, it is very cold, but its also really special.
