See answers below:
Question #1: The most important mechanical part of a vehicle is the engine. The engine is the heart of the vehicle. Other parts are important too and have their own roles such as cooling systems and charging systems.
Question #2: Technically, the vehicle won't run without an engine. You can do without the other stuff like air conditioning system.
Question #3: The vehicle is working and there's a lot of benefits as long as your vehicle is working fine.