I do it using both of the ways you described, without either of the problems you described. KDE Plasma 5 on X11.

Not sure if this would help, but I've found it nice separating my open programs between workspaces. Game on one and other programs in side tabs. Can quickly access the panel by switching workspaces.

Does take a bit of self-adjusting to how it works on Windows as you need to organize in your mind which program is in which workspace.

From my experience at least, some games are just like this, more often than not when using proton. I don't really have this issue that often after switching to i3wm. Can you switch to a different workspace? Is your game in fullscreen? It usually works better for me in windowed borderless. The benefit of using i3 is that you can run games in standard windowed mode, and let i3 fullscreen the application, or just switch workspace entirely. Hopefully this helps.

Try going into a menu first.

i usually have one window open in the background and alt+tab into that, or if not, i hit windows key and it lets me type in the menu and i can launch another program from there. alternatively you can press win+d and it'll throw the game out of sight. wondering if alt+esc would defocus the game, am not near my pc so i can't test that yet. or how about swapping desktops? (ctrl+alt+right/left)

If you can't fix it you could try running the game inside Gamescope, it can help with weird alt+tab issues.

I do not experience the behaviour with the cursor, but when a game in fullscreen doesn't play nice, I move it to the second workspace (with a keybind) and then quickly switch to the first workspace with another keybind. Both are easy to remember and as fast as alt + tab.

No idea where is that in Linux Mint (Cinnamon) but you probably have a "focus follows cursor" setting somewhere. If you manage to disable it, you will only switch focus by clicking.

Personally I use bspwm and instead of switching between windows, I switch between workspaces. I also have focus follows cursor setting enabled and it's essential for my workflow, however it can be disabled.

You can't have screen scaling on and not run in the native resolution for the game window. Might be your issue.

centering the mouse on the game

Maybe I'm missing something, but.. why is this a problem? You can just like, move the mouse to where you want it?

I can't replicate this issue on mine. I recently somehow broke Gnome (again) during an update, so I finally went back to a tiling WM (awesome this time). I start the game in a separate desktop, and then when I need to do things outside of the game, I just change to another desktop using the keyboard shortcuts. Pretty sure this is possible in all WM/DEs on Linux.