The train do stop but the amount of stops depends on how long you can go for your trip In my case I had only 6 weeks for the Mongol Rally so we drove every day

The train do stop but the amount of stops depends on how long you can go for your trip In my case I had only 6 weeks for the Mongol Rally so we drove every day 14 hours in the car I'm not sure how long it takes to do the Trans-Siberian route, but if I will do it I will make sure I have enough time to have few days in nice places along the way.