Hi,
The flat battery, what to do depends on it's age, if it is more than say 4 years old then it is probably no longer reliable but it may still recover so shall we say less than for years old and it should recharge and be reliable. 4 years or more , if you want peace of mind replace it but it's still worth a gamble if you are happy with the risk that it is likely to do the same again.
The next question is why did the battery run flat?
Is there a drain on the battery with the car parked, check current drain from battery with everything turned off?
Is the charging system not working in some way, with the engine running I would expect to see about 14 to 15 volts across the battery?
Lower than that and the alternator is not working or there is a wiring problem , engine connection to car body or battery connection to the car body are the most common problem in my (limited) experience.
I would think the the flashing odometer is trying to tell you the electronics in the car are unable to function because the voltage is far too low.
See:
https://www.fiat500owners.com/thread...nything.88825/
What is being described is to disconnect the battery, discharge any residual power left in the system to clear the memory, then reconnect the battery.