One of the reasons a textfield is used instead of a textarea is to limit the number of characters typed. The chatterbox can only handle messages up to about 255 characters long. When you hit this limit, the textfield stops accepting more input which tells you to shorten your message or break it in two. I don't believe you can put a limit on the number of characters entered into a textarea.

If you let your mouse hover over the name of the person who sent the message, the time that the message was sent should be displayed. You could also look into Other CB Clients as recommended at the bottom of the chatterbox. Some of those clients (like my favorite, the Java chatterbox client) display the time next to the message.