Maybe this will provide some interesting info to other people in this long but interesting project, as I wished I had all that info in just 1 place...

Desktop : i8700k, r9 fury, dualoot : win10 and archliux
Client : raspberry pi 4, 2go
Wired network. Hdmi to my Home Cinema Amplifier which feeds the video signal in a projector (1080p)
raspbian from https://www.raspberrypi.org/software/ (4.7.1)
To avoid the prompt whether to execute Steam Link when double-clicking the desktop icon : edit /etc/xdg/libfm/libfm.conf and add the following line under the [config] section:
(reboot required)
Fixed ip :
Open the file /etc/dhcpcd.conf and replace:
In the config panel : enable ssh
To remove the annoying ssh password warning (I kept pi/raspberry)
I always use mc to edit files via ssh. But default colors are terrible (can’t read certain parameters in the editor). Download solarized.ini from https://github.com/iwfmp/mc-solarized-skin and put it in ~/solarized.ini then (you can’t edit mc config files with mc : they are overwritten when you exit)
Go to skin and change default to /home/pi/solarized.ini
Ctrl+o to save, ctrl+x to quit
Readable, at last !
There’s no 5.1 profile, so (after many hours of searching…), I went to output to « Digital Stereo ». On my Amplifier, I can set some processing « Dolby pro logic » which simulates some surround effects by treating a stereo signal. Not a true 5.1, but better than just stereo.
I had audio dropouts : when the pi starts playing a sound, the amplifier must configure itself, so it starts playing after 2s and then disconnects. I tried this trick in my ~/.config/pulse/default.pa

But, same result.
Finally, the update2 of this (https://unix.stackexchange.com/questions/362223/short-audio-playback-is-muted-requires-warming-up-or-secondary-audio-in-backgro) solved it : it continuously sends an inaudible sinus to the speakers.
apt install sox
Create this file : ~/.config/systemd/user/continuous-silence.service
And enable it with
Success !
I plugged a real push button between pins 5 and 6 (gpio3 and gnd)
Then saved in /home/pi/shutdown.py
In the control panel>wait for network (otherwise you get some « can’t update » message boxes)
I created ~/start_perso.sh
I put steamlink twice because I sometimes make a mistake and press B, which brings me back to the desktop. With that, it relaunches steamlink a 2nd time.

Create ~/.config/lxsession/LXDE/autostart with (it will disable this one : /etc/xdg/lxsession/LXDE-pi/autostart)
On desktop : remote play : enable hardware encoding just for intel igpu
On client : quality : beautiful (was blocky with balanced with some snow on a lego game), stereo, hevc enabled.
Good quality, fluid. I’m sometimes under the impression that some button pressed are slightly delayed (or even skipped), but I might be because of the projector lag (or me).
I have 2 dragonrise homemade arcade sticks

In steamlink client : config>Gamepads : I was able to map them correctly (except : I can’t map the dpad, but maybe it’s because I mapped the stick to analog and it won’t accept to output analog and dpad at the same time).

I changed in windows control pannel, but it always displayed the old name on linux. Editing
~/.config/Valve Corporation made the trick
Force hdmi output even if tv is off
In /boot/config.txt:
As I had big black borders on the projector :
# uncomment this if your display has a black border of unused pixels visible
# and your display can output without overscan
disable_overscan=0

# uncomment the following to adjust overscan. Use positive numbers if console
# goes off screen, and negative if there is too much border
overscan_left=-40
overscan_right=-40
overscan_top=-40
overscan_bottom=-40

All this was not sufficient : to be able to get a picture if I switch on tv after the pi4, I also added a « dr hdmi » box between the pi and my amplifier. Maybe you can do it just with the proper config on the pi, but after hours of searching, this is what worked.

There’s a problem separating the inputs (can’t play versus : the 2nd stick is not wroking). The trick is :
1) enter « arcade mode », up to the character selection, then go back
2) you can now play versus with 2 gamepads.

I’m rather pleased with the system (but I’m just beginning using it).
1) I wish I could find a proper solution for a true 5.1 output (dolbyprologic simulation is not bad, but just for the sake of having the best out of my hardware...). It should be as simple as selecting a 5.1 output profile in pavucontrol. The hardware is capable (as libreelec works fine with 5.1 output).
There’s some discussion here: https://steamcommunity.com/app/353380/discussions/6/1743353164095764346/
I spent a lot of time trying that : on a retropie install (without desktop), I was able to configure alsa via a .asoundrc file and get speaker-test -c6 -Dsurround51 output to my 6 speakers. But, with steamlink, my rear speackers were aways silent.
On a raspbian, the .asoundrc file is always deleted (even if you change its permissions) : they have switched to pulse audio, so they don’t want to mess up with alsa anymore. So this config is outdated for raspbian at least.
2) I tried to get pegasus frontend to work (as emulation seems to be only non-x) : could never launch it (event with flatpak). I abandonned : I will only use steamlink interface (and maybe launch retroarch from it).