As the good Samaritan I am, I have tried Dawn of War 2. With standard settings, it would try to start, but fail. What I did was: Right-click the game → Properties → Compatibility → confirm the option → select Proton Experimental (let it download stuff).
With Proton Experimental (as mentioned in the ProtonDB page), the game works - I watched the cinematics, selected difficulty, was able to move through the map. Cloud sync is not working, it shows an error, that is also mentioned in the ProtonDB link.
Though, as you are missing the garuda-inxi output, I do not know your config. I am on full AMD build, you might have further issues with Nvidia. Also, I do not have the other two games. And I did not and will not try multiplayer.