I have the same exact problem. Its either a coolant pump of some kind or the power steering pump running (you can feel the vibrations there). So I guess the battery is shot?

 

i had a similiar issue with my car not starting and guages dropping to zero. it was a dirty/bad connection on the battery. once i cleaned the terminals and tightened them back down mine started up fine. not sure what the humming was, but my fan would stay on for a while after driving the car hard. this could drain the battery. turned out i had a leak in the coolant and a thermostat going bad, which caused the car to heat up and the fan to come on a lot. i changed out the coolant and thermostat and now the fan doesnt remain on for long periods of time before.

not sure if you experienced the same thing as me, but just making a suggestion that it could be more than one issue that causes your symptoms.

good luck!

 

Hello, All ..hoping you can help - Have a 2006 Mini-Cooper S, only 49K on her. (Great condition) Noticed recently when starting her can hear a small almost electric revving noise up to 2000 rpms. After that a constant humming noise - no 'rev' noise turning, stopping at lights, etc. When park her and as turning off ignition can hear that small revving noise, again. It does stop after ignition off. Any thots, Pls, would appreciate. TY!