1) The drive does not depend on the power from the Airport Extreme, I have the drive plugged into it's own power.
You did not understand what I said above. My Lacie hard drive has its own power supply. Yet, it will not appear in AirPort Utility if I connect it to the USB port on my AirPort Extreme........unless.......I use a powered USB hub. You may have different results depending on the drive that you are using and how old your AirPort Extreme might be.
2) After opening the AirPort Utility, and clicking Edit - I do see the drive in the Disks tab
That is encouraging.
3) Yes, I have restarted the AirPort Extreme and my Mac
Got that, thanks.
4) It is currently using an IPv4 address (screen shot attached)
Got that, but Apple uses IPv6 for some internal communications between the AirPort Extreme and your Mac, so that is an area that we will probably need to look at.......but......first I would like you to see if the drive will appear in the Finder by doing the following:
Click the smiley face Finder icon on the dock
Click the Go menu at the uppermost top of the Mac's screen
Click Connect to Server
Enter the following in the address box afp://10.0.1.1
Click Connect
You may be asked for the AirPort Extreme admin password.....unless.....you have set up a separate disk password for the drive in AirPort Utility, in which case you will have to try the disk password
If things are working, the drive icon should appear on the desktop after a few seconds and also appear under the Locations menu on the left panel of the Finder window
Post back on your results