Don't know about Synaptic, but you can log the output of any command on the command line using the tee utility. For example,
apt-get dist-upgrade | tee /tmp/apt.log
will copy the output of "apt-get dist-upgrade" to /tmp/apt.log .
In general, you can copy text to the clipboard by dragging your mouse with the left button pressed. Then you can paste the highlighted text in another terminal or application by pressing the middle mouse button. This doesn't work in all applications, though.