Intuos 3 Drivers for Mac OSX Sierra (10.12)
After searching all over the internet, I found a few people who have the same problem here.
They said to use Wacom Driver 6.3.8-2 (macOS 10.7 - 10.9).
This driver can be found here.
Alternatively, you can find the link to this driver by:

Navigate to Wacom's Driver Search Page here
Search for &quot;Intuos 3&quot;
In the MacOS driver box, select the &quot;Older Versions&quot; link
Scroll the list until you see Driver 6.3.8-2 (macOS 10.7 - 10.9) and download it
Un-install any previous non-working versions of the wacom tablet driver and install the newly downloaded version

After I installed it, I was able to see the Wacom Tablet on the Preferences Pane and edit the preferences to map the tablet to my screen size again!
My pen and mouse work as they did previously, but the original issue with my mouse (clicking) now appears to be a hardware issue. I will go buy a replacement mouse from eBay.