1) Firebug is useful, if you're familiar with it and detail oriented.
2) Click on the agent number above the chat box (if still visible) and drag down and if your reflexes are quick enough you get the text. LOL
3) Here's the fun one: (at least on our installation)
Click somewhere inside the "Mut Alert Sound" text and drag UP into the text you want to copy. A little bit of magic then occurs and it suddenly highlights ALL the text in the box. Then when you click Copy, you get all of it.
4) Yep, I think a "copy" button would probably be a great feature request. Perhaps you should sponsor it ...