I use the Mayflash ps3 dongle and, if it's the same as mine, it can handle 1 controller through Xinput and 4 controllers through Dinput with a switch on the side to select what you want. You'll never have to worry about controllers reordering, as the first one you activate is always first player. IMO, it's the most versatile setup and the Dpad is excellent for retro console games.
Yes mine has the XInput/DInput switch and I was aware that XInput only allowed one controller but if the DInput is 4 rather than 2 as I had thought, then even better. Regardless, it's a great little product and there's something satisfying about putting some really nice cordless controllers to use that you already have on hand rather than stuffing them into a box full of dusty Rock Band instruments in the closet (or worse, a landfill).
I use Raspberry Pis, but if you aren't comfortable with Linux already I'd say stick with a PC. For most things it is going to be way simpler to use than a pi (unless you use a pre-built image.)
Hard to beat it for size though.
As for wireless... eh, I'd say get longer cords, but I think that a matter of personal taste. Any setup with Bluetooth will work with wireless controllers. I'm not a fan though... I even prefer wired controllers on the PS4.
Yes, there are definitely positives for the Pi. Just recently I was playing around with mine (Pi 3B) and after spending a few hours trying to sort out some issues I was having with screen resolution and finally solving them, I was very impressed with how nice of a little retro console emulator it was.
A key benefit of the PC route - at least for many people - is that there is usually some level of pre-existing understanding on how to use a PC that isn't there with Linux. Not only that, but decades' worth of internet information to assist with setting up the emulators on PC. The Pi community is growing but it's much newer and less fleshed out, and with Linux, doing something as simple as copying a file can feel daunting if you aren't familiar with the OS.
To be fair, a PC-based system in no way guarantees you won't run into any frustrations and/or a learning curve with the software. Especially if you're using an old-ish PC and Retroarch, there's a pretty good chance you're going to need to tinker under the hood to get smooth operation and games running at full speed. For example, I spent a few hours trying to get Retroarch to run smoothly on my LG 4K TV. It just didn't want to work right unless I forced the refresh rate to 30Hz, which caused other problems. I moved the system over to an older 1080 TV and it runs like a champ.
As for cordless controllers, I guess the thinking is that wireless inherently introduces lag. For me, the cordless option is just too clean and convenient to pass up and I am perfectly OK with the level of lag I'm getting (i.e., I don't even notice it).