Would a PS4 Controller Work With These Current Console Emulators?

Would a PS4 controller work with a RPCS3/PCSX2/ePSXe emulators?

Yep.

As long as Windows recognizes the controller you're golden.

There's a couple of ways to do this. Steam has native support so you can use remote play through steam, you can get DS4Windows which is a little program for it with drivers. This works from Win7 through 10.

Which one would you recommend?

Either works, but I don't use Sony controllers. It's easiest with a 360/Xbox One controller because Windows will recognize it right off the bat (Windows 7 may need a driver from Microsoft which takes 2 seconds to do, versions of Windows after that simply work)

While we're on the subject, would a Nintendo Switch Pro/PowerA controller work with an emulator for NES, SNES, Game Boy/Game Boy Advance, and Nintendo DS/3DS? Gamecube maybe?

Or a Sega Saturn controller solely for Kega Fusion?