@mitu hello again, I was finally able to connect my DOBE-TNS-19252C after many attempts. In case anyone is having the same problem, the adapter Mayflash Magic-NS-2 did the trick in my case. I connected the controller to the adapter and the adapter to the Rpi, all via USB, and it worked like a charm. I was able to configure the input of my controller without a problem.

Regarding the problem with Joycons, and in case it is of help, I’m using clones and not the official ones. I tried connecting them while booting and afterwards, both without success.

Thank you so much for your help!

See this forum post for running games directly from a physical media. I don't think lr-pcsx-rearmed supports physical media though, for PS1 you need the Mednafen PSX core.
As far as Gamecube/PS2, I don't think the emulators in RetroPie (Dolphin/PCSX2) support reading the game from physical media.

As I am still in mock up testing...

I thought why not just try... worst case it fails and I have to get a new encoder.

So I have taken out the good old soldering iron and moved the J1 jumper from next to R2 as it came delivered and moved it next to R1 Using same middle contact.

Now when plugging into the USB it boots up with the Green light "on" being in digital mode, the mode switch if desired still lets me change state.

I made a video comparing the encoders one left as delivered starting analogue with red led, and one modified by relocating the jumper (before modifying the 2nd one).

I have done exactly that myself because for some reason I couldn't get retropie to run on top the Lite OS. Long story short, I like it. You can set boot order to run Retropie first and you got a desktop environment as a bonus when you exit RP. I have yet to find any disadvantages to that... Space efficiency maybe, if you have a smaller SD card?

Are you playing it under NTSC or PAL settings? If the former, maybe try under PAL.

@Ataru22 Yep, the gun(s) play like to be in the lowest number USB Ports on the PI, its easier to configure that way. I have two guns plugged in like that and have a USB hub for all the stuff, after those and it all works fine.