Why don't (electrically charged) particles act on themselves? As per the Maxwell-Gauss equation, an electron modifies the electrical field around it. Therefore it should act (through an electric force) on itself. Now obviously this force would be directed towards the electron itself and its vector would be along $\overrightarrow0$, but since the distance from the electron to itself (neglecting quantum considerations for now) is 0, the force vector can be expressed in cartesian coordinates as :
$$
\overrightarrow F=\frac{e^2}{4\pi\varepsilon_0r^2}\overrightarrow0=\frac{e^2}{4\pi\varepsilon_0r^2}\cdot0\hat{e_x}=\infty\cdot0\hat{e_x}
$$
Now this raises an infinity, which heavily suggests that this expression for the electric force is wrong in this context, and maybe there would be a more correct (set of) formula(s) from quantum mechanics that I do not know (please tell me if that's the case).
But it's also an indeterminate form — is there a mathematical argument that waives it? Or is the act of considering a force applied by the electron on itself nonsensical in the first place?
My current guess for an argument is that symmetry considerations clearly make it so that the electron in this experience should have no "preferred direction", so there should not be any net force. But this could also mean that the electron does interact with itself, and that the forces it applies on itself are always cancelling each other out.
Maybe this is more of a "problem approaching" issue than a "fundamental physics" one. But please let me know if this is related to anything interesting (even if it's probably not).
 A: In a nutshell, classical electrodynamics isn't suited to the description of self-interaction. In order to lift that restriction, you need to add a quantum ingredient to the soup, in other words switch to quantum field theory.
Although it isn't the whole story, a good picture could be the following.
As you get closer to the electron, you start to see the structure of its electromagnetic field, so you start to see photons. In particle physics, "getting closer" means "going to higher energy", so at some point you have enough energy to create electron-positron pairs. QFT provides the tools to describe the complex structure of interacting particles, which is precisely a description of the particle interacting with its own field.
A: Point particles do self-interact, sort of.  Remember that the description of an electron as a “particle” is a semiclassical simplification which sweeps an awful lot of modern physics under the rug.  In quantum field theory, “an electron” is a quantized excitation of a spinor field associated with a particular mass, charge, and other quantum numbers.
In the context of quantum field theory, when we say that “the electron is a point particle,” we don’t mean that some zero-size analog of a sand grain exists somewhere to be located or not.  A better interpretation is that the electron is structureless.  No matter how closely you look at an electron, there is no new interaction which switches on so that “an electron” is no longer a good description of what’s happening.  This is different from atoms, which can be driven into excited states and eventually separated into electrons and nuclei; from nuclei, which can be driven into excited states and eventually separated into protons and neutrons; and from protons and neutrons, which can be driven into excited states like the delta or lambda baryons.  (Protons and neutrons can’t actually be separated into quarks, for reasons which are too complex for a parenthetical.  But we have reasons to believe that quarks, like electrons, are structureless.)
The length scale of a quantum-mechanical interaction is set by the de Broglie wavelength of the particles/fields involved.  High-momentum interactions probe short-distance physics.  The shorter your interaction distance, the more high-energy effects start to leak in.
The first of these effects is called the “vacuum polarization.” Quantum electromagnetism is mediated by virtual photons, which can spend part of their time as virtual electron-positron pairs.  At high energies, these virtual pairs become more and more important.  The overall effect is that the fine-structure constant,
$$
\alpha = \frac{1}{\hbar c} \frac{e^2}{4\pi\epsilon_0}
$$
is actually a “running constant.” In low energy interactions, $\alpha\approx 1/137$.  By the time the relevant length scale corresponds to the masses of the weak vector bosons $W$ and $Z$, the effective electromagnetic coupling is a little stronger, $\alpha\approx 1/127$.
If you insisted on thinking of this in terms of classical electromagnetism, you might say that the electric field is slightly stronger than the $1/r^2$ prediction very close to “the electron,” due to vacuum polarization, a kind of self-interaction.  But if you push closer to this “core,” then the weak interaction becomes unignorable, and classical electromagnetism is no good to you any more.
For non-relativistic quantum mechanics, you can get pretty far by equating the electron’s probability density $|\psi^*\psi|$ with its charge density.  So an electron in an $s$-wave orbital has the largest charge density at the nucleus, but the charge density is relatively uniform within roughly the Bohr radius.  A uniform charge density distribution does not suffer from the singularity that you’re worrying about.
A: Within classical electromagnetic theory, there are two options.
If electric charge carried by the particle is distributed on different points of space, then there is self-interaction due to this charge, because parts of the charge distribution act on different parts and in relativistic theory, net force due to this need not be zero. This is known to be so in EM theory when such extended particle moves with acceleration; due to acceleration of charge, there is induced electric field that acts against its acceleration and decreases effective inertial mass of the charged particle, and there is also the radiation damping effect, approximately given by the Lorentz-Abraham force, which act in direction of change of acceleration.
If electric charge is concentrated in a single point, then the above description with self-interaction does not work consistently, and self-interaction has to be excluded to get logically consistent theory. There is no consistent classical EM theory of point particles that includes self-interaction.
For electron, we do not know whether it is point particle or has some small dimensions. There are estimates based on experiments with scattering and g-factor of electron saying that if electron has non-zero size, it is smaller than 1e-18 m.
