With help from an H&D modder, Jakub, I have finally found how to solve the issue.
Apparently, the Editor retains the Properties window opened in a specific set of coordinates but seems to move away the rest of the interface: the Editor *is* working but the interface is displaced. Jakub was kind enough to show me a workaround, which I'm now sharing so that others with the same issue can now access the editor.
1) Install H&D Deluxe, and choose Full Game + Editor on setup.
2) Once everything is installed, open the Windows Registry (Click on Start > Run > then type "regedit" without the quotes)
3) Head over to HKEY_CURRENT_USER / SOFTWARE / Insanity 3D system / hdedit.exe (as seen on the attached image)
4) Right-click on the Window Position value. You will find the values "FF 7F 00 00 FF 7F 00 00".
5) Replace the above values with "FA FF FF FF 00 00 00 00".
This will "reset" the window position for the Editor. Simply close the Registry and run hdedit.exe.
Hope this helps anyone who had issues with the Editor!