Cora is not going to let the players run the show. He is in complete charge. That said, one of the things that makes him such an effective manager is that he is willing to listen to his players and he takes their input into consideration. At least, that's my impression of how things are based of what I've read and seen.
Pitchers who are really good can usually throw to any catcher and not miss a beat. However, if it gives a pitcher more confidence and comfort to throw to a certain catcher, and said pitcher and catcher have developed a relationship to the point where they know each other extremely well and are very effective together, why mess with a good thing?