That last part was a bit tricky.
If I recall correctly, you need to store the full log in a String (or even better, a StringBuilder) field within the Game class. Every time the user takes an action that adds points, you need to update that String to include the new message.
Because the data is all stored in the Game class, then even if you go to look at another game and come back to the game, you can set the TextArea's text to match the log text that's associated with the game currently being viewed.
Hope that helps!