If the probability of a point (photon) hitting another point (electron) is zero why do they collide? If the probability of a point (photon) hitting another point (electron) is zero why do they collide? To have a probability greater than zero almost one of them should be not a point. Correct me, please if I am wrong.
 A: In popular presentations of particle physics you often find the statement that photons and electrons are "point particles", while protons and other composite entities are not. However to call a photon or an electron a "point particle" is quite misleading unless you immediately add that we are dealing with quantum physics and a quantum physical "particle" is always spread out over some range of position and momentum.
When you see Feynman diagrams it looks a lot as if one little point-like thing comes along and absorbs or emits another little point-like thing at a vertex in the diagram, but this is wrong. The lines in the diagram usually represent states of well-defined momentum and energy, which means the position of each entity (e.g. electron or photon) is totally spread out, so these states are as far removed from "point-like" as they could possibly be!
In practice what happens is that you have states that are intermediate: neither completely spread out, not completely focused at a point. Such states are called wave-packets.
The basic example of a photon interacting with an electron is the process called the Compton effect. If the incoming photon and electron were each focused in such a way that their wave-packets never overlapped then indeed they would not interact! The Compton effect is observed when the wave-packets do overlap.
A: The photon interacts with the electron, but it does not collide with it.
You can think of an electron as having a cloud of virtual particles, which are basically excitations in the quantum EM field. The quanta of this field are called photons

Take for example, this simple interaction between 2 electrons:

As shown with a Feynman diagram.
Explanation of the diagram
The 2 solid external lines on either side of the diagram are the electrons and the internal line in the middle of the diagram (the wavy line) is the virtual photon. This photon is being exchanged between the 2 electrons. This can be thought of as an exchange of momentum between the electrons.

Answer
Since every particle that has an electric charge has a cloud of virtual photons, the photon exchanged between the particles gets absorbed into the cloud of virtual particles of the particle that it was propagating towards.
And since the EM field is not in a fixed position or a point, but rather it is a field that expands out from the electron throughout spacetime. The photon interacts with this field rather than the election itself. Therefore, the exchanged photon gets absorbed into this field and the particles travel on their merry way, away from each other (in this case).

This whole process can be calculated mathematically with the S-Matrix or using the Feynman rules if you want to work smart.
A: 
To have a probability greater than zero almost one of them should be not a point.

In classical mechanics, when one models charged particles as point particles, their center of mass (CM) being the point, and the trajectory of the CM of one particle aims at the CM of the other, there will be an interaction that will exchange momenta and change the  (x,y,z) location of both  particles, because of the Coulomb force between them. The probability of interaction  depends on the initial  conditions, the probability of a head on collision depends on the volume of the mass of each particles.
Your puzzlement belongs to this classical  framework.
The theory of quantum mechanics developed when it was experimentally and in observation discovered that the classical theories were not enough in order to model data and, important, be predictive of new. The main impetus for a new theory came from the photoelectric effect, black body radiation and the spectra of atoms. It started with what is now called first quantization (FQ), the wave equations that modeled and predicted the spectra of atoms (Schrodinger, Dirac)  , imposing on the solutions  of the wave equation postulates, extra axioms, in order to pick up those solutions that modeled the data. The solutions, interpreted as the probability of a particle being measured at (x,y,z,t) agreed with all observations.
Then data became accumulated in higher energy particle scattering, higher than the electron volt energies of atomic observations, and it was necessary to extend the theory to what is called second quantization (SQ) , which is the level of calculations with Feynman diagrams. SQ keeps the postulates of FQ  The QFT for the point particles of particle physics is the Standard Model (SM), there the particles in the table are assumed to be zero point particles, axiomatically.
The other answers to this question are given within this SQ theory, which has been very successful both in modelling and in predictions.
In a similar way that in classical mechanics particles are treated as points with a mass about their CM, in the SM  particles are treated as points with a field around it. It happens to be a field described in probability space, i.e. many measurements at a given (x,y,z,t) with the same boundary conditions have to be made in order to see the predictions of the model, but it works.  Because the model is successful we describe the elementary particles as point particles .
