I'm not quite sure why, but I think the chat controls would feel more natural on the right, rather than the left.
My idea, as seen below, would be to move the arrow thing to the right side of each chat "bubble". The controls would appear on mouseover, like now, but we could star or flag right from there (the most common tools) without having to have a second window to do that in. The arrow would let you open the window to access other controls, permalink, etc.
On your own posts, which you can't flag or star, you could change the controls to edit/delete, as those are the most common things to do on your own posts.
And this is what your post would look like:
Of course, if you make this change, then the occasional time stamps would have to be moved.