Temperature is a property of the system, it units are Kelvin. Heat is energy transfer by virtue of a temperature difference, its units are Joules. Temperature and heat are not the same thing. Heat is what we call the energy when it crosses the system boundary. Practice makes perfect.
Air is less viscous than honey, but the forces are still there.
Of course due to gas's low density, changes in kinetic and potential energy are often small compared to changes in internal energy. Whether or not something is a function of the state of a system or path dependent depends on its behavior in the world around us. The equations we are working with were developed over many years to model the world around us, and they do a pretty good job. The energy from the motor eventually all ends up as internal energy. After the fan is shut off, eventually viscous forces will cause the gas in the box to slow down and then stop moving.