None of these things are consistent day to day -- save for the fact that I don't know what you mean by "players or coaches call the shots." Have no idea what that means. So, I'll offer some of what goes on in the post-game interviews.
The manager speaks first in the televised press conference.
On the road, the TV reporter usually asks the first question.
At home, one of the print reporters, like Hummel, will ask the first question, but not always. It's random. It is sometimes as simple as who gets a microphone stuck in their face first.
The clubhouse is open to the media after the manager leaves it for his press conference.
Some players prefer to wait for all of the media to arrive. Some players will speak immediately to the reporters who have chosen to skip the manager's televised press conference.
Reporters ask questions, players answer. The wheel turns.
Not all of the questions are broadcast as many of us wait for the television cameras to leave the scrum so that we can ask more detailed questions or get answers that we would like to only appear in print. Also, just like you or I, players answer differently when there is a camera in their face and when there is only a print reporter with his notepad.
The manager is similar. Not all questions are televised because there are some questions that are best asked when the manager wants time to think of an answer or to elaborate on an answer, or there is news that we want to explore without the camera present. This happened the other day when Alex Reyes as optioned. There were no questions about that because why scoop ourselves by broadcasting it before we can have a story up?
Some players don't come out. Some players leave. Some players decline comment. It really depends on the day.
Yesterday, I went to the visitors clubhouse, participated in Green's post-game, waited for the Padres beat writers and TV crew to ask their questions and then asked my Cardinals-specific one (that's being professional polite to my peers), and then went into the clubhouse, where Hosmer was at his locker. I asked if he had a moment for an interview. He said he wanted to finish sending a text message, and then he did. I waited. We talked. Other reporters joined around me as I asked questions. And then when that was done, another reporter and I waited for about 20 minutes more to spend 2 minutes talking to Manny Machado. He questioned my ability to ask a question at one point -- and that gave me the ending to my story. QED.