Currently using an X11 system, on an AMD GPU; the window manager is FVWM because I'm a nostalgic old git.
I use two screens, and most games tend to full-screen on one.
Had decent enough results with Proton via Steam on many titles. A few of them needed to be explicitly tagged "don't draw a frame around the full screen window" in the FVWM config, and I had a few where movies did that "show a test card instead of video" but no biggie.
I've recently had two harder nuts to crack. I'm using two games with Lutris: The SNK 40th Anniversary Collection (it was $20 cheaper on GOG than Steam at the time!), and Genshin Impact.
Both of them play fine, so long as you keep the mouse within the full-screen capture area. But if I leave the window (say, using a keyboard combination or pulling the mouse outside the capture area), the games go blank.
SNK shifts the black box somewhat off of its original position, and I think Genshin just goes blank.
I experimented a bit with SNK's "wine configuration" options in Lutris.
"Automatically Capture the Mouse in Full-Screen Window": This reduced accidental leave-the-screen problems, but still had failures if you used a keyboard command to switch windows.
De-selecting "allow the window manager to control the window" causes the window to turn into a weird Win95-esque "mini taskbar icon" instead of going black Pressing the "restore" and "maximize" buttons resizes it to near full screen but retains an ugly Win95 style title bar. Once you restore it in that mode, it's actually well-behaved-- you can move the mouse in and out of the window without it breaking (it seems to freeze when you move the mouse away, but that may be intentional) But still, the weird titlebar and it not working that way until you first "freeze and unwedge it" sucks.
Genshin, at least sometimes, could have its black box minimized and restored and come back to life. I've yet to try the Wine tweaks there.
I suspect the common theme might be that the games are trying to deactivate themselves when they lose focus, but not doing so gracefully. ISTR Genshin on Windows would minimize itself if you switched to another task, and I haven't tried SNK on actual Windows. I'm wondering if there's some unified fix that tells the game it's running in a single screen and when the mouse leaves, it just stayed there. There seems to have been some sort of "cage Wine apps in a virtual desktop" feature, but it seems to no longer be supported.
For the video problem, it might be codecs; try using Proton-GE if it's in Steam or use Wine-GE if not. (IIRC, Steam will often convert the videos and give you the converted ones in the shader caching if necessary. But those outside of Steam, and sometimes a few still in Steam don't have that workaround.)
For the main issue you're having, try running those games in gamescope, which itself is a compositor with a bunch of neat tricks. In this case it'd make sure to not lose the focus of the game even if the gamescope window loses focus. It can also optionally force windowed or fullscreen modes, upscale (even with FSR1), and lock the framerate.
Changing settings in the game itself between fullscreen or borderless (borderless should usually help with the focus issues) may help too, if the game has that setting, but then you'll probably hit the borders issue due to FVWM. (I don't know if you'd get the fullscreen unredirection optimization in fvwm. That could be a reason to pick one or the other for you too.)
You're probably hitting a few edge cases by using FVWM versus a more modern environment like GNOME or KDE, but to be fair I've seen the focus issue happen before on a game on running through Heroic on GNOME with more than one monitor before. FWIW: I don't remember seeing the issue in games from Steam. (It probably depends on the game itself, however.)
Gamescope seems to have done a good job of taming the SNK games. Genshin.... seems to have fixed itself. For a while it was in a weird state where the game worked but the launcher beeped furiously though the PC speaker, like I was sending beeps to an xterm. Now it seems fine. I do feel like this is a lot more black magic than I'm used to with Linux-- I actually had to reboot to get to a consistent behaviour-- but non-native games do tend to play fast and loose.