I installed this game on linux through Lutris app. The game installs ok but does not start. Only the black screen appears and that's it. Any help is useful. Thank's !

I don't know about that Lutris thing, but it works with Wine. You can make a script or a launcher that does the following :

cd ~/.wine/dosdevices/c\:/GOG\ Games/Medal\ of\ Honor\ -\ Allied\ Assault\ War\ Chest/
env WINEPREFIX="/home/user/.wine" MESA_EXTENSION_MAX_YEAR=2002 wine MOHAA.exe

Tested & approved on Wine 4 & 5