Would output signal affect input signal when the opamp is saturating?
Yes, in any case where there is feedback that touches the input (many cases such as inverting amplifiers, Sallen-Key filters and so on) and the source impedance is greater than zero (all real cases), you will have some effect if the output saturates. 

The simplest example is probably an inverting amplifier- say you have a gain of -1 with two 5K resistors, and a source impedance of 100 ohms. 

When the amplifier is not saturated the input "looks" like a 5K resistor to ground, so the voltage from the source (taking the internal resistance into account) is simply 5000/5100 times the unloaded output voltage. So it's attenuated by ~2% but no distortion if the amplifier output is following exactly. Now if amplifier saturates, it appears as a 10K resistor to the saturated output voltage, so the slope of the output increases a bit (to ~1% attenuation) and you have a point of inflection and some distortion. 







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