How is it traveling (Trans Siberia) in Russia - China in December?

I'm planning to do the Trans-Siberian trip (St. Petersburg to Beijing) in December. I have roughly around 12 to 15 days to spend and I'm traveling on a shoestring (using pocket money & student allowance :P teehee). I heard tickets and accommodations are cheaper and always available during the winter. Do I have to pre-book everything or is it possible to just walk in and get tickets / accommodation? Can anyone suggest me where to stop, for how long and what to do? And how is it like crossing the country's border? Do I have to get off the train, fill out form and get my passport / visa check at the immigration?

Asked by J Tristan via WorldNomads.com