It's part of an electric motor, I'm not sure what from though- I'd guess perhaps not automotive, the size suggests something like an electric drill or similar.
Are there any numbers stamped or engraved on the casing?
You will need to check each and every bush, steering or suspension joint for play - they will need to be absolutely perfect as any play will be multipled due to the lift.
Also checking all mounting points for steering components etc.
No, it's not faulty. If your system blows plenty of cold air then the compressor works. If it isn't coming on when unless the temperature is set low then something else is wrong, the thermostat is a likely culprit, but not the only possible one.
Ethanol is hygroscopic, meaning it absorbs water, if there was moisture in your tank it could have mixed with the fuel and contaminated it. Mixing in fresh fuel may help the situation. However, if fuel was the problem there would probably be misfiring in more than one cylinder, however there's no harm in trying.
I'd still be concerned about spark, ...
The symptoms you report are those of a partially or fully discharged battery. Ten minutes' running is insufficient to even partially charge a discharged battery.
To troubleshoot, you must start with a fully charged battery installed in the car. Either put the battery you have on a suitable charger for enough time, or replace the battery you have with a new, ...