ZDoom

Just a heads up as a Linux user myself, make sure everything is properly capitalized. Linux does allow you to have "folder" and "Folder" - or any variation there of in the same folder. So if you have a file for an example: ".../pwads/theWad.WAD" and you type in ".../pwads/thewad.wad" Linux will think you're looking for that exact file path, not ".../pwads/theWad.WAD"

Having just tried running GZDoom in my terminal, it works as intended.

As for the gzdoom.desktop issue you can make a new shortcut yourself. I myself use KDE and any system generated shortcut is bound to root. Just make a new shortcut - the new shortcut would have you as the owner and not root.

They don't. That is one of the locations in the FileSearch.Directories section in the ini by default, so it is a directory that GZDoom will search if the file name you gave doesn't resolve in the working directory but otherwise nothing special about it.

Update: I have found a solution. It seems like the WADs you want to play with need to go in .config/gzdoom.

Those are the things that work, which leads me to believe that either the wad is broken or "/path/to/the.wad" is wrong. Look at the console, there's probably some information near the top when it tries to read the file. (Particularly with the terminal commands you posted.)

Things I have attempted in order to run a PWAD on Ubuntu:

Test the WAD in SLADE:
Doesn't work.

Drag the WAD file onto GZDoom in the Dash:
Nothing happened.

Drag the WAD file onto GZDoom in Nautilus:
Nothing happened.

Run gzdoom /path/to/the.wad:
Only the IWAD opens, this with TNT as the WAD is designed for.

Run gzdoom -file /path/to/the.wad:
Only the IWAD opens.

Right-click the WAD and click "Open with...":
GZDoom is not on the menu.

Attempt to edit gzdoom.desktop in order to put GZDoom in the "Open with..." menu:
"File system is read-only", even with sudo.

Linux nerds, what are you doing that works?