AnFI,
You may be interested in this quote from the EC135 training course notes:
"After attaining self sustaining speed of 50% N1 the start relay disconnects the ignition circuit. From this point on, combustion of the fuel/air mixture continues without the aid of outside ignition".
The compressor is spinning at 54,000rpm and the power turbine at 44,000rpm - when the fire goes out, it goes out quickly and the fire stays out. There is no auto re-light.
For a restart in flight, the N1 must be below 17% and the engine main switch must start from the "Off" position. If the switch is moved from "Off" to "Idle" before the N1 is below 17% then nothing will occur until that level is reached.
When a gas turbine is shut down correctly, there are several processes that occur to ensure that the engine is not damaged, injectors do not become coked or bearings do not get carbon'd. If an engine suffers a "Flame out" then it is withdrawn from service until it has had an engineering inspection.