Stop/Start might be down to the battery level being low, especially if the car is only used occasionally or for mostly short journeys. The system will deactivate if the battery level is low to ensure the engine keeps running to keep it charged.
Unfortunately Fiat have also fitted a battery monitoring system on the alternator which shuts off when the battery is at the nominal level, to avoid overcharging and prematurely wearing out the battery. I say unfortunately as this system doesn't seem to be particularly good, on my old 2015 500x it killed a battery in 2 years, the level dropped to 6vdc after 10hours (and a very long drive from France to mid UK). The RAC chap advised the battery monitoring system was at fault but my dealer refused to accept this... I gave the car back shortly afterwards as I'd lost confidence in it getting me to work and back each day ( a 2hr round trip and a mix of urban and dual carriageway).
With regards to the engine management light, this should be logging a fault for them to read so I'd be very surprised at this. Have you got a Fiat specialist nearby, might be worth them checking it out too?
2015 500X Pop Star, 1.4