There were no trains running all the way across. Sometimes one car would be switched in Chicago, but everyone else changed trains. ("A hog can cross the country without changing trains, but you can't.")
Most transcontinental trips changed in Chicago, but there were some that switched in the South.
General passenger train arrangement:
Baggage & Mail cars
Lounge cars (first class but not sleepers) probably with a platform on the last one.
Dining cars among the sleepers and lounges, possibly among the day coaches.
Private and official cars tacked on the rear.
Sometimes there might be groups of cars that would be cut off for some intermediate destination not on the main route.
There might be extra express boxcars/reefers at the front.
Numbers of cars would vary with the marketing of the train. All-Pullman trains didn't come in until the 20s or 30s. (Confirm?)
A mail train might be almost all mail cars with a coach or sleeper tacked on the back.