Two stage op-amp design with compensation ideas
The capacitor across the feedback resistor adds a zero in the loop. That is to say that it increases loop gain but it also adds phase, improving stability. This capacitor also adds a pole at a higher frequency which has the potential to cancel out the stability improvement of the zero. So the capacitor value must be carefully chosen so that the loop gain gets down to unity before this higher frequency pole is reached.

The compensation capacitor's value would be chosen to give the desired phase margin, usually for unity closed loop gain, but maybe for a higher closed loop gain for a decompensated op amp irrespective of how the external circuit is configured. The designer then has the option of including a capacitor across the feedback resistor to further improve stability if desired. This capacitor is sometimes included for the dual purpose of reducing high frequency noise.

EDIT

You can vary the current in both the input and output stages together by varying Ibias. To vary the current in just the input stage insert and vary a resistor between the source of M5 and Vdd.

The current in either side of the input stage should be kept balanced (equal) or distortion will result.

At a simple level of thinking, slewrate is set by the current in the input stage or the current in the output stage, which ever is smaller.This is because it is these currents which charge the compensation capacitor.

For example, if there is 20uA total in the input stage and 35uA in the output stage then the slewrate equals (assuming a 100pF Ccomp) :-

SR = dv/dt = i/C = 20uA/100pF = 0.2V/us

EDIT

Open loop gain is determined by the transconductance, Gm of the input stage and the impedance of the compensation capacitor at any particular frequency.

O.L Gain = Gm.Xc

Gm does decrease as the input stage’s differential amp’s tail current decreases, but this shouldn’t reduce GBW because as Gm decreases (due to reduced tail current) you would decrease the size of the compensation capacitor (increasing the O.L gain back up) to achieve the desired phase margin of somewhere between 45° and 65°.

EDIT

That op amp design could do with an output stage on it: