Suppose the player closes down the app (ex. close web browser for HTML5 app and close app for exported Android/iOS), is it possible for the player to still receive some text notification?
I'm not really sure how you see it with HTML5 app. When user closes the browser then there is no place where message should be sent to. In that case you can send him a message if he has some account in your app (on your server), and he will be able to read it once he logs in to the app. There is plenty ways to do that but I think this is not what you want.
However if browser is opened and player is in your app then you can send him a message using WebSockets.
Regarding mobile version, there is no built in plugin to do Push Notification in C2, but it is possible to do that with CooconJS and some manual scripting if I remember well. Just Google "construct 2 push notification" there are some information related to Ludei.