There were indeed no events when I tested with evtest.
I completely removed the keyboard again, rebooted and started Pixel. I again paired the keyboard, and made sure it worked in Pixel. Then I rebooted into ES again, again no keyboard..... some kind of weird inspiration made me go to the Retropie Config (you know, where the setup.sh is started) and, much to my surprise, I found that after a few button pushes the keyboard started working! I have no idea what went on and why it worked, because apart from opening setup I did nothing differently from the first time. After leaving the config I started evtest again (via the remote console) and actually this time around the events were showing up! The keyboard actually started to work!
Soooo happy now, as after adding the proper event source to the moonlight config I actually have full keyboard support there as well!!!
Thank you again for your help and patience @mitu !
After much searching, I found this link https://github.com/goodtft/LCD-show/issues/278 for an issue with an LCD display I tried to use years ago. Apparently, it updated my /boot/cmdline.txt file to put ttyAMA0 in my console, instead of the default serial0. Swapping it back to console=serial0,115200 and rebooting, and now bluetooth works again!
While waiting I installed Recall box to try if it works. It's same there.
However... I found old Saitek Pad (PS2700) and it works like a charm. I decided to try xBox Controller via USB and it works like a charm.
Trying to connect via Bluetooth does not work at all. Will try to ssh and play with BT then, maybe console will tell me something.
Since I can't pair my headphones also, I think it might not be pad problem, but bluetooth.
About SteamController. There is problem with mapping of d-pad. It recognizes only swipe in one or 2 directions. Like Analog Left and Analog Up. So its more like Analogue for emulator than D-Pad.
But I think i can pass on stemController if XBox and Saitek works. I'm a fan of D-Pad. Touchpad as D-Pad is not good on SteamController.