Firstly, consider the 'game history' button as compared to the 'analyse board' button. Both open the same screen, but the analysed board function does it in a new window, avoiding the need to 'return to game', which is often slow. Perhaps both should open new windows?
Point number 2: Message log (more significantly really)
This would be much nicer if it opened in a new window thanks to the time spent going back / forward to the game screen and also in that you might want to refer to one message in the log while writing another on the game screen.
Point 3: tooltip the last message over the current one? Maybe tricky, but it would be well cool: no real need to open the message log at in most cases then. (I imagine there isn't really room to leave the last message up on the screen as well as a new one).