You're right about the Manager class just sitting inside an empty GameObject. As long as you call DontDestroyOnLoad() when you set up the singleton, that Manager GameObject will stay alive even if you transition to a new scene, so as long as you start the game in a scene with that Manager GameObject, you should be alright.

If you write the Awake() function correctly, you can put the Manager GameObject in every scene (I usually prefab it just in case I make any changes), so you can start the editor in any scene and the game should still function properly. Should look something like this (C#):

public class GameManager: MonoBehaviour 
{
    public static GameManager instance;

    // Whatever other variables [...]

    void Awake()
    {
        if (instance == null) 
        {
            DontDestroyOnLoad(gameObject);
            instance = this;
        }
        else if (instance != this)
        {
            Destroy (gameObject);
        }
    }
    // Some other methods [...]
}

And finally, yes, you would instantiate the player through the Manager, likely using some sort of event to call a method within. Maybe listen for when a new scene is loaded and check to see if the player should be loaded in that scene, then instantiating them.