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There are three ways that a colliding photon can cause an electron to be ejected from the nucleus : via the photoelectric effect, the Compton effect, and pair production. Does the initial kinetic energy of the ejected electron depend on the type of interaction that took place? Is it perhaps greater for CE than PE etc.

I think no. The kinetic energy of the electron is proportional to the kinetic energy of the incident photon, and the exact method by which the photon was generated does not matter. So that as far as ionization effects are considered, the dominant photon interaction is not particularly relevant.

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I think it does, during PE for example the photon leaves all its energy to the nucleus for the electron to be ejected, in this case I think you could find a proportionality relation between the energies of the two particles. However, in the case of CE, the photon leaves a part of its energy and it continues its trajectory (with angular deviation and energy loss), the incident photon could generate several free electrons before being absorbed or escaped. The simplest case is the pair production where the photon loses 1.022 MeV for the masses of the electron and the positron and it gives the rest of its energy in equal amounts for the two particles.

That being said, I don't think you could generate a generalized relation linking the energies of the incident photon and the liberated/produced electron. I hope this helps.

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