Fiat 500 Electrics, ECU, Warnings & Lights Problems

New battery fitted,car started and ran but the next time i tried to start it everything was absolutely dead,i jumped off mym wifes car and it started and ran but after disconnection my car was again totally dead.

when i hit a bump or go over speed bumps my front and back windscreen wipers come on by themselves help.

Air bag light will not go out Fiat can't find problem and say airbag is working but local council will not let me run this vehicle with light still on as it is a taxi any ideas.....thanks for your time eddieok.....