How the FADEC knows the maximum power that he can make at 40 degrees?It has a database?
Yes or a method of calculating the power that will be produced at a given temperature and barometric setting.
What the FADEC do when we put the levers in TOGA?
It sets the power according to the temperature you have entered.
How the FADEC does knows the power that need to put, if we don’t put neither temperature?
You have to enter a temperature.
What happens if we put a flex temperature lower than the outside temperature?
That is impossible because you are now demanding more thrust than the engines can provide. If you need that performance you are overweight. Therefore you change the aircraft's configuration, the runway you are using or reduce the payload.
It has to do with EGT limit?
You use reduced thrust to reduce the engine's operating temperature and resulting wear and tear on the engine and airframe. You are unlikely to be near the operating limit of the engine.
PM