Here are some reasons you would want a negative supply:
If the op amp is not a rail-rail type you will need a negative supply for the input and/or output to go to zero volts.
If the op amp is a rail-rail type then you will need negative supply if the signal goes below zero volts (such as an AC signal with average value of 0V) and you want the DC value of the signal to be preserved.
If the amplifier does not have a negative supply voltage, it can not output a negative voltage. Many amplifiers can not input a negative voltage if they do not have a negative power supply voltage. Therefore, the role of a negative supply voltage is to enable the amplifier to take in and put out negative voltages. The amplifier chip can not make negative power supply voltages for itself.
I see that you posted but the picture didn't come through. Probably stuck in the self protecting features of the site. It denies some content by people with very little history here until a moderator approves it.
Because a coupling capacitor will *always* introduce frequency response artifacts.
A dual supply (both positive and negative) has a nice zero volt reference point available to amplify signals against. "Zero" can be used as both an input an output reference/return point.
A typical dual supply op amp has no idea the supply is a dual supply. It will work to full specification with a single supply if signals are referenced to a suitable point.