A first good step could be to clean out the auxiliary port on the Gameboy with isopropyl alcohol preferably 96%+. After that then inspecting the inside of the jack for damage. After checking for damage if you see any or if you continue to have problems then it might be worth while to look into checking the solder connections form the Board to the Aux jack and then possibly replacing the jack.
All I can recommend is checking then reseating the connections to the screen. These being the one near the camera up top and the one in the middle right area. If you still run into problems it is possible that the replacement screen is faulty.
I would recommend replacing the battery and making sure that that’s not the problem. Be careful and make sure that you have the right model controller as the battery does not fit the JDM-040 model DualShock 4 controllers. A replacement battery is sold by ifixit here: DualShock 4 (JDM-030 and Earlier) Controller Battery And a guide to replacing it can be found here: DualShock 4バッテリーの交換
The connection within the failing side might be faulty, perhaps SteelSeries has a temporary warranty or the company you bought it from does and you could have them replaced. If so you could give them a call and trade in the broken pair for a new one.
I am unsure if you can upgrade your CPU but I have heard and seen Evidence that Windows 11 isn’t fully optimized for gaming yet, I would stick to Windows 10 unless you have a specific reason to move up to 11.
I am by no means an expert but I would start by making sure that my Power cables leading to the board are okay, then possibly checking the Motherboards Power in for shorts and then moving on to using a thermal camera to look for excessive heat areas, then checking the chips with the most heat output for shorts.