Game launches on the secondary monitor and on windowed mode, fixed it on kde by right clicking the X11 icon>More>Move>Fullscreen

MANGOHUD=1 gamemoderun %command%
When playing native the mouse had a small offset. Switching to proton fixed the issue for me.
Native also worked fine but the mouse had a small offset.

I got slightly better performance by using Proton experimental. Otherwise the game worked flawlessly from start to finish.
Works perfectlty
MANGOHUD=1 gamemoderun %command%

Works fine on both proton and native versions, proton had better performance though
DXVK_FRAME_RATE=60 gamemoderun mangohud %command%
At max settings fps might be a bit bumpy so better cap it 60

Every time I open the game, I need to alt-tabbing to recognize the window and commands.
Played with Proton-Experimental [bleeding-edge] in Very High settings (SSAA off), 1080p fullscreen with 60 FPS stable.
Incredible and immersive shooter post-apocalyptic game.
Just install and play!

Crashed once
Game ran smoothly with Steam settings set to default Proton Experimental. No Proton version was forced in the game properties. Only hiccups were in the loading screens between stages, but did not affect the game play. Only a single crash, and I did not lose any progress in the game.

It's not perfect but it is good and works well
Some flickering from light sources. Fairly minimal, in my experience it was worse with proton over native
Crashed maybe twice throughout an entire story playthrough.
I did experience some minor issues while playing which has been listed. Overall the experience was a good one. Just keep in mind that you do miss out on some graphical settings running native over proton. Not a big deal for me though.

Funciona perfecto, sin inconvenientes.
It runs really well. Only problem I had was that I could not play the game with 2 monitors, but that is an issue of the game, not linux.
Runs perfect on high settings with capped 45 FPS. Played the whole game near end experienced some FPS dips to 30 for several seconds.

It just works

gamemoderun mangohud %command%
Game is fullscreen by default. It caps the fps to 60 while fullscreen. Had to switch to borderless window through config file:
r_fullscreen off
Game doesn't have in game fov setting so had to change it in config file at:
r_base_fov 64.
Config file can be found at:
/home/$USER/.steam/steam/steamapps/compatdata/286690/pfx/drive_c/users/steamuser/AppData/Local/4A Games/Metro 2033//user.cfg
Fullscreen can't be turned off in-game

The native Linux version seems to always launch in windowed mode on Wayland, and there are no resolution or fullscreen settings at all in the native Linux version for some reason. I had to switch to Proton in order to be able to play in fullscren. Flawless after that. The Linux port seemed to run well too, if they fix the fullscreen issue it would be okay.
Playing native works fine with stereo speakers and a basic mouse, but you'll need Proton if you have more audio channels or mouse buttons.

Played start to finish with no issues on native port.

gamemoderun %command%
Both the Native and Proton version work great.

FSR ON // SHARPENING 5 // RES: 960x600 // SSAA: OFF
Change some buttons
TDP: 12 W
Perfect
prime-run %command%

Worked perfectly, no tinkering needed.
I rarely had slight FPS drops, nothing to worry about.
Loading screens would freeze up, otherwise fine.

Fullscreen issues in native version
TDP 10--12

May work native, definitely works proton
Native version on my laptop (integrated Radeon graphics), models would extrude and lighting glitch, making the game unplayable. Native on my desktop (RX 6750 XT) did not have this problem. Proton version runs with no noticable problems.
Steam cloud does not download between Windows and Linux saves, but the versions have save compatibility.

Depending on your setup there might be some issues that need to be fixed, but then it plays very well.
gamemoderun %command%
When running on Wayland (sway), the game could not properly use the 2560x1440 resolution of my monitor. Trying around with gamescope didn't help me either. But on Xorg all worked as expected, so I'd recommend doing that for this game.
When using Proton experimental or Proton GE the mouse in menus would get stuck (in game the mouse worked, so it's still mostly playable). Using Proton 8.0-3 (or earlier) instead solved that issue.
Extra tip: Look at https://www.pcgamingwiki.com/wiki/Metro_2033_Redux#Skip_intro_videos to disable the lengthy intro cutscene.

can only play in 1080
native version only ran on second monitor
Mouse was stuck on experimental and ge8-4
this game does not deserve the gold rating it currently has.
Works wonders on Linux. I recommend using proton instead of using the native Linux version.
DXVK_ASYNC=1 gamemoderun %command%
Didn't find any problems with the Proton version of this game, it runs normally and doesn't explode your pc like Windows does
Artifacts appear when using the Linux version, lots of stuttering too
Game runs just like on Windows, with Proton-GE obviously better (Proton-GE commands go brrrr), but you can play normally as well, it's good
gamemoderun %command%
Sometimes it slows down, specially in loading screens, but I just have to wait some seconds or if in the game, I pause and then continue.
The game works fine.
Tried multiple versions of Proton and the native Linux version, and none work
Tried the settings recommended here and elsewhere for 40hz and disabling tesselation and PhysX
Steam Cloud does not work for this game, despite what the Steam store says
Frequent crashes, rendering the game unplayable
Despite the incorrect Steam Deck compatibility rating, this game does not run on the Deck. It has constant crashing and freezing, and it also doesn't work with Steam Cloud. I've uninstalled and reinstalled, deleted Proton files, tried different versions of Proton, etc., and nothing works. The worst part is the game works just long enough to suck you into the story and then crashes, and it crashes at different places every time, and if you reinstall you'll lose whatever little progress you already made (since Steam Cloud doesn't work), so you have to start the whole game over just to see if it'll work correctly this time, which it doesn't.
I've played through this on Xbox, and it's amazing. If they fix the problems for Deck (or re-rate it to show its incompatibility), I'll give it 5 stars.