Hi,
sounds like one ore more plug connections are faulty. Generally, the FIAT wiring concept is known as - let's say - tiny.
When some years ago my Doblo seemed to make efforts to join the show business (light show in the dash, strange use of the outside lighting) i unplugged every cable connection i could reach in order to clean them. While doing so i found a broken wire which i was able to fix.
After that operation the van worked like a charm up to now with one exception (The oil pressure sensor has communication problems with the ECU. Mom appears a bit lazy.) When turning the ignition twice on before starting everything is fine.
My Stilo is producing a quite similar behavior.
FIAT eleccatricity is a bit special.
Good luck