Thanks for the additional information.
1 Ignore the running on for the time being. Might just be the back EMF from the radiator fan.
2 Voltages as above.
3 Electrical connections: you can just look and wiggle, or you can break, inspect, clean (cloth and fluid up to light abrasion) and reassemble.
4 I don't know my way around all all the sensors or a logical way to test them...
Good luck
Jonathan