Pb with trigger Group

Try pushing your hammer down farther into the trigger catch
it is not down far enough

The front of the trigger catch is hitting the safety
also why is the housing shiny where the trigger hits the safety

In "sleepling" position, it's normal.
The safety doesn't hit the trigger catch.

The shiny place come from the safety, it's hit the trigger frame.

When I push the trigger: it's not enough to liberate the hammer

ok with the hammer locked down pull the rear latch back it should move freely and the spring housing should go up slightly.

yes, that's work like wrote.

ok last idea for me does the back of the trigger hit the housing

Have you tried to disassemble this TG? I notice in the picture there seems to be a lot of clearence around the trigger pin. I'm talking about between the pin and the housing. If you have an accumulation of clearence (wear) it seems possible for there not to be enough movement in the trigger to relaese. Most of the problems I have seen with TG are the result of people trying to "improve" these mechanisms. You said this one is from storage, so I guess you have another one to compare it to. There are several good videos online showing how to disassemble and reassemble these things, but if your not comfortable doing it you may want to find someone to help you. I don't have any idea how hard that would be in France.