Common mode feedback in 2 stage amplifier

[Chinmaye]

I have designed a fully differential 2 stage op-amp with resistive CMFB for both the stages. This is to build a switched capacitor Multiply by 2 amplifier (SC MX2). Although the Gain, Phase margin and bandwidth were satisfactory for differential input of the amplifier. The amplifier was not giving required output for certain inputs. Initially i felt it might be because the common mode gain of the 2 stage amplifier was greater than 1 for few common mode input voltages. Later, I read about connecting a decoupling capacitor in parallel to the CMFB resistors. I tried a few values for capacitors randomly and the SC MX2 started working fine. I still do not have clarity on couple of concepts.

Does a CM gain of greater than 1 ruin the stability of CMFB? What can the max tolerable value of CM gain, PM and bandwidth? How does connecting decoupling capacitors in parallel to CMFB resistors help? How can one choose the value of the CMFB capacitors?
TIA