Don't put up with using Proton/WINE to shim windows games for your Steam Deck or Linux distro.
Make use of a combination of amazing open-source projects to get more performance out of your game and system.
Thanks!

You can get Luxtorpeda direct from their Git page (https://luxtorpeda-dev.github.io/)
But if you are lazier like me you can use a tool like ProtonPlus to install compatability shims in the right location somewhat automatically (e.g. https://flathub.org/apps/com.vysp3r.ProtonPlus).
Select the right Steam installation (You might have 2 if you have the Flatpak and the DEB/RPM)
Through the GUI select Luxtorpeda
Note: Luxtorpeda does need updating from time to time, you only need the latest version installed though so feel free to uninstall the older version after installing the newest
In the Properties of the game, go to the Compatability tab, enable Compatability mode then open the drop down to select Luxtorpeda. The first time you run the game it will download the Linux-native files to your installation directory.
Get the list here; https://luxtorpeda-dev.github.io/packages
If the save files are kept in the game installation directory, yes
If the save files are kept in the Documents folder that is passed to your Home folder through Proton, yes
If the save files are kept in another directory in the wineprefix of the Proton compatibility folder, no but you might be able to save them using PCGamingWiki to find the save file locations (https://www.pcgamingwiki.com/wiki/Doom_3)

Linux Installer For Linux Gamers is an amazing project that appears to have a bunch of rare setups https://liflg.org

They have a Doom 3 + RoE all in one update. I found that using these files in your Doom 3/RoE directory alongside launching the game with Luxtorpedia gives the best performance

Visit https://github.com/liflg/doom3_1.3.1.1304-multilingual_x86
Either:

• Select "<> Code" and then select "Download ZIP"
  
• Use the command line in a directory of your choice e.g.
  mkdir ~/Games/doom3/d3linux cd Games/doom3/d3linux git clone https://github.com/liflg/doom3_1.3.1.1304-multilingual_x86

Take all files in the folder and paste them into your Steam D3/RoE installation directory

This looks pretty sick, will see what happens when we merge it
https://github.com/dhewm/dhewm3

Fullscreen using either Luxtorpeda or - resulted in 1FPS. Running Luxtorpeda first, then overwriting the game files with the contents of - resulted in nice smooth gameplay looking like the original with no loss of save progress.

Don't let me forget

But Luxtorpeda has worked for me every time on the SD so why would it give up now?

To dhewm project team, Luxtorpeda project team, liflg project team, and all the shoulders of the other giants that they stood on to get us here. Donate a dollar or more to them when you can, they give up their lives for our enjoyment.