Bofink, the answer is yes, you can connect your laptop HDD while your desktop computer is running, that will not be a problem. It will take a few seconds for the PC to recognize the drive, and then probably a new "drive conected" message will appear. Then you can use Windows Explorer and copy your files.
It seemed by the picture that installing and unstillaing is simply plug-in, plug-out...easy to do.
Now just to be a little more specific, you have to plug in the second drive after windows is already running. If you boot with both HDDs installed, this could cause windows to fail to boot. It wont mess with your data, but you wont be able to get into windows. But there is no problem at all pluging in a second drive with an OS installed in a already running machine.