Here are real events relating to the last page of the pdf link you gave:
Fig.1 This bubble chamber picture shows some electromagnetic events such as pair creation or materialization of high energy photon into an electron-positron pair (green tracks), the Compton effect (red tracks), the emission of electromagnetic radiation by accelerating charges (violet tracks) (bremsstrahlung) and the knock-on electrons or delta ray (blue tracks)
Photons are invisible in bubble chambers as they interact only with direct collisions with electrons, called Compton scattering, or pair production in the field of a nucleus. Charge particles turn in the perpendicular to the plane magnetic field; this allow us to measure their momentum and charge and the ionisation of the tracks allows the identification of masses.
At the lower left of the picture, there is an electron( identified by its ionisation) which loses energy into a photon, and the photon pair produces some centimeters away, an electron positron pair.
in the middle right side, we see a positron that loses energy into a photon and the photon kicks an electron from the atoms of the chamber, this is a Compton scatter. This corresponds to the diagram in the last page of your link, except it has been reduced to one dimension. In reality there are two dimensions, because the photon gives part of its momentum/energy kicking at an angle . The following is the correct diagram kinematically:
It should not be surprising that classical scatters and particle scatters kinematically are the same, because momentum and energy conservation hold both classically and quantum mechanically. It is the probability of interaction that is different in the microcosm of elementary particles to the billiard ball particle scattering. In simple scattering experiments the kinematics are not different ( except that special relativity holds in the microcosm).
Edit after edits in question.
I think the image , from a real experiment, answers whether a photon can hit an electron or not.
Now you ask:
What if I put the electrons behind a double slit apparatus, and treat individual photons as particles? Based on this "compton scattering," it's possible for the photon to be deflected any which way. I could claim that the diffraction pattern observed in the double-slit experiment is due to compton scattering, among other factors. Prove me wrong!
The difraction pattern of individual photons, even when sent one at a time, is a direct result of the quantum mechanical nature of the photon. The solution of the boundary conditions imposed by the two slits gives a probability distribution that displays an interference pattern. Even though electromagnetic interactions viewed as Feynman diagrams are similar, it is the boundary conditions that determine the probability of scatter, and two slits is different than two particle scatter, the fields are different and the solutions are different.