To make it stay in "drop-down mode", click the little "tack" icon next to the drop-down button, when you have it in the "drop-down" mode. Now it will stay open when Classic loses focus.
If you want Classic to be "always on top" or "auto-hide", set that option under the View | Visibility menu.
Set to auto-hide to have it go away when you complete a QuickPaste.