First off, I just created this topic for anyone experiencing these two issues. If you have upgraded your PS4 controller to a newer DualSense, in order for it to work, you need to enable native imput through settings and run at the same time DS4Windows emulator in order to emulate an Xbox 360 controller. The game will recognize your DualSense now, but be warned, pressing the touchpad will cause the start and share buttons to stop from working. Restarting the game fixes this. Luckily, the touchpad is not necessary during the game as the inventory buttons gets replaced by the share button, and you can still use the keyboard to press those buttons. Using a DualShock 4 profile with DS4Windows will make the touchpad work again, but the vibration won't work for unknown reasons.

Second, if you're running the latest version of the game on gog, you probably noticed the infamous achievements glitch which affects most of gog users. All of your achievements will be blacked out in the main menu, and all of them will pop up at once each time you sit at a bench. You can just turn off the achievement notification at the main menu, or if you care about them, modify a file. As stated here https://www.reddit.com/r/HollowKnight/comments/137c055/psa_how_to_reenable_achievements_on_the_gog/ you need to rename GalaxyCSharpGlue.dll within Hollow Knight\Hollow Knight_Data\Plugins\x86_64 folder. Just add a .bak extension, and after the next launch, the achievements will be good as new.

I managed to get my DS5 controller working in DS4 mode using DS4Windows 3.3.3 (the latest version as of this post) and fix any touchpad issues with a couple tweaks in DS4Windows:

1) Disable touchpad slide (I enabled the "Start with Slide/Scroll Off" option).

2) Rebind Left/Right/Multi/Upper Touch to "Touchpad Click" instead of the default mouse clicks.

3) Use HidHide (included with DS4Windows) to disable the real DS5 controller so only the emulated DS4 controller is detected. I'm unsure if this step is necessary, but it probably doesn't hurt. I'm definitely gonna forget to unhide it later, though...

The GalaxyCSharpGlue.dll achievements fix worked like a charm, too. Thanks for the post.