Sounds like two different issues (but maybe not).
The electric fan not only runs when the A/C is on but will also come on if the air temperature sensor (top right corner of the fan shroud) reaches setpoint and turns on the cooldown relay. This can happen even when the engine is off and the key removed. Since the fan no longer runs after engine shutdown, it may be as simple as the engine bay temp is not high enough. Or it could be the sensor or the cooldown relay are bad.
The beeping (screeching?) noise may be fuel being bled back to the tank through the fuel pressure regulator (which it should NOT do). This can be checked with a fuel pressure gauge installed in the fuel rail.
OTOH, if the electric fan is not working, the fuel may be boiling after engine shutdown. The pressure increase may then be forcing fuel past the regulator resulting in the beeping.
Series 1 Euro 1750 GTV
Series 2 USA 1750 GTV
Series 3 Spider Veloce