Updating S7 Edge using Odin: stuck in Download Mode's "Downloading..."
Okay so I eventually managed to both unbrick my phone and install Oreo. Here is how I did it.

Unbricking the phone:


When the phone is off, press Power + Home + Volume Up (yes, up)
Don't worry about the weird messages, just wait a minute
A menu will appear with several options, one of which will be to factory reset the phone. Use the volume keys to go to this option, and then the power key to select it
As if by magic, your phone will reset and then be unbricked, and the OS will load normally


Installing Oreo - what I missed was:


Enabling OEM Unlock in Developer Options in the phone's settings
Using the latest version of Odin - v3.13.1 - it appears that anything older won't work
(possibly) Running Odin as administrator on the PC


I recommend this video tutorial to install Oreo: https://www.youtube.com/watch?v=mdSIyk7lkd8

Good luck!