I'm a little confused with the OP here.
The situation may be different on newer generation aircraft, but on the A320 series, the only time FADEC will shut down an engine is during ground auto-start. In flight, FADEC will NEVER shut down an engine. It may, under certain failure conditions, command idle and there are some faults (not directly FADEC related) that may cause a flame-out but the decision to actually shut down an engine rests entirely with the pilots not the FADEC system.
As for whether to re-light? Follow your company/Airbus SOP.