Why is there a capacitor in parallel with this fan? What is the benefit?
The capacitor is there to reduce radio emissions.  Things have to be switched as a function of rotation angle to make most motors run.  Those switching transients can cause voltage spikes, which can emit RF interence.  The capacitor shorts out the high frequencies and keeps their currents local.  It will also slow down transient edges that the FET sees.  Too high a dV/dt on the drain of the FET can cause problems.