You shouldn't have any trouble doing the Trans-Mongolian independently. I did the Trans-Manchurian (which goes directly from Russia to China, avoiding Mongolia), and I bought tickets as I went, with no previous reservations. No problem. Of course, the Trans-Mongolian is a more popular route, but since you're travelling in January, the low season, you shouldn't have any trouble.
Lake Baikal was the best part of the trip, when I went. Don't miss it!