Question

Which parameter of an op-amp become significant for a fast varying ultra low input voltage signal?

Answer 1

For ultra-low input voltage it is better to use so-called "Rail-to-Rail in" op-amps. The point is that with very low voltages the noise will be sow huge that the op-amp can't work effectively at all. Rail-to-Rail amplifiers have very good dynamic range that they can amplify signals almost from -V to +V.

For high frequencies the op-amp must have high cutoff frequency because the gain decreases when frequency increases.