As other posters have alluded to, there are many trains that do all or part of this route and they don't all have the same facilities, so if you're dead set on having a shower then make sure you check this before you buy your tickets for a particular train. Though if you are going to break up your journey along the way (thoroughly recommended) then a shower on the train probably won't be a critical feature.
I did this journey (but east to west) in the winter, and the snowy landscapes were something special. In fact I would probably have felt cheated if I hadn't seen Siberia in white :-) However, inside the train it was roasting - T-shirt and shorts conditions.
Would definitely second the idea of learning some Russian, as the majority of passengers are not tourists (though, depending on how you buy your tickets, you may be lumped into a compartment with other tourists). You will have plenty of time for reading - I found that "War and Peace" fitted nicely into the stretch between Krasnoyarsk and Yekaterinburg.