chitgoks
1) Yes, you have to buy a ticket from Schaffhausen to Stuttgart if I understand correctly your question.
2) You can show the STP and the ticket for Germany after the departure from Zurich HB. If the staff change later after the border, you will be asked to show the ticket for Germany again.
3) 07:13 is the arrival time in Schaffhausen and 07:16 the departure time. You can stay in the train, but other people have 3 minutes to get off or get on. This train is also available for local trips and not only from Zurich to Stuttgart.