javafx using objects from MainController or other Controllers in proper Controller class

You are creating a controller instance “by hand” with

MainController mc = new MainController();

@FXML-annotated fields are initialized by the FXMLLoader when it creates the controller for you as part of the process of loading the FXML file. Since the controller you created is not the controller instance created by the FXMLLoader, its @FXML-annotated fields are uninitialized (i.e. they are null), and hence you get a null pointer exception.

You can get a reference to the controller created by the FXMLLoader by calling getController() on the FXMLLoader instance after calling load().

If you want one controller to communicate with another, then pass the reference to one controller to the other controller, by defining appropriate methods in the second controller:

public class ConnectionErrorController implements Initializable {

    private MainController mainController ;

    public void setMainController(MainController mainController) {
        this.mainController = mainController ;
    }

    // ...

    @Override
    public void initialize(URL location, ResourceBundle resources) {

        infoLabel.setText("Connection lost, please try again");
        tryButton.setText("try again");
        exitButton.setText("exit");

        tryButton.setOnAction(new EventHandler<ActionEvent>() {
            @Override
            public void handle(ActionEvent event) {

                WebEngine webEngine = mainController.getContentPaneController().getVideoWebView().getEngine(); // 1
                ToggleButton playButton = mainController.getControlPaneController().getPlayButton(); // 2
                if (mainController.testInet()) {
                    mainController.play(webEngine, playButton);
                } else {
                    // obviously you can now do something better than the "public static field hack" here:
                    MainController.exist = false;
                }
                tryButton.getScene().getWindow().hide();
            }
        });

        // ...
    }
}    

Assuming you are loading the second fxml file in a method in MainController, you can then just do something like:

public class MainController {

    // ...

    public void showConnectionErrorWindow(String fileName) {

        FXMLLoader loader = new FXMLLoader(getClass().getResource("path/to/ConnectionError.fxml"));
        Parent root = loader.load();
        ConnectionErrorController connectionErrorController = loader.getController();
        connectionErrorController.setMainController(this);

        Scene scene = new Scene(root);
        Stage stage = new Stage();
        // etc...
    }

    // ...
}

Note that there are much more elegant ways of solving this problem, such as passing the ConnectionErrorController a callback function (in the form of a lambda expression) to process the call to play(...), which avoid the tight coupling between the ConnectionErrorController and the MainController. However, as you seem to be new to Java, this simpler approach might be more suitable.

See Passing Parameters JavaFX FXML for more information on passing values to controllers.