The complete list of participating countries is here:
https://de.wikipedia.org/wiki/Gemeinsamer_Internationaler_Ta...
However, an Eurail pass will most likely be cheaper and you will enjoy greater freedom in chosing your connections:
https://en.wikipedia.org/wiki/Eurail
I suspect the Moscow -> Saigon leg to be a bit more complicated, but there are travel agencies specialized in offering exotic train rides.
From Beijing it shouldn’t be too difficult to find a single link to Hanoi. And from Hanoi there is a single sleeper that runs to Ho Chi Minh (I’ve taken this one in the other direction back in 2008).
So if I’m not mistaken (or out of date) Moscow to Ho Chi Minh should be at most 3 connections (Ulan Ude/Chita, Beijing, and Hanoi).