To copy/paste text in an Adobe Flash Paper document, you must use the "text select I-beam" just to the left of the search text entry window. Select that tool and then highlight the text you wish to copy. Use the keyboard shortcut CTRL-C (hold the Control key down and press the C key) to copy the text. You can then paste it into another program using CTRL-V or the Edit Paste function of the other program.
UPDATE: This function is available with Flash Player 9.0 or earlier versions, but will not work with Flash Player 10.