About Self-Energy/Self-Potential Energy I am studying Special Relativity. When calculating the Electrostatic Energy of Point Charges, there is self energy which is infinite due to the interaction between the charge and the Coulomb potential produced by itself. The author said this should be ignored as it has no physical interest. It also says it means there is a fundamental limitation for classical electrodynamics (dimension needs to be >1e-25m). 
Can you tell me if this is solved in more advanced physics (e.g. quantum electrodynamics) and if this is well understood in the community or it is still a research topic? Thanks a lot!
 A: In the framework of Quantum Field Theory (a class of theories, to which Quantum Electrodynamics belongs), the concept of elementary particle is modified dramatically.
In Quantum Mechanics, you don't have point particles anymore. Elementary particles are in a way extended. They are represented by complex-valued functions over space, called the wavefunctions. The value of this function at a given spatial point is connected with the probability of observing the particle at that point.
Quantum Field Theory takes this even further. Elementary particles are modeled as quanta of relativistic fields. Thus, there is no place for a point particle model in QFT. And the infinite energy problem does not arise. Well, except that it does :)
Actually, you have to deal with a whole lot more of these nasty infinity problems in QFT. These have been classified into infrared and ultraviolet divergences, based on their origin (infrared divergences originate from large-scale fluctuations, whilst ultraviolet divergences are present in almost all QFTs with interactions and originate from short-scale fluctuations). A systematic approach for dealing with these divergences in the framework of perturbative QFT has been developed. For more details on this, see this answer of mine.
I hope this helps.
A: 
When calculating the Electrostatic Energy of Point Charges, there is self energy which is infinite due to the interaction between the charge and the Coulomb potential produced by itself.

This is true if we begin with the formula that was introduced for continuous charge distributions
$$
E_p = \frac{1}{2}\int d^3\mathbf x \int d^3\mathbf y ~ K\frac{\rho(\mathbf x)\rho(\mathbf y)}{|\mathbf x-\mathbf y|}
$$
based on the formula for potential energy of point charges:
$$
E_p = \frac{1}{2}\sum_i \sum_{j}{}^{'} K \frac{q_iq_j}{|\mathbf r_i-\mathbf r_j|}.
$$
The prime next to the second sum over $j$ means the sum over $j$ is to be done for $j$ that are not equal to $i$.
For point particles, it makes no sense to use the first formula, point charge has singular density that makes the integral nonsense. The appropriate formula is the original one, where no divergence occurs.

The author said this should be ignored as it has no physical interest. It also says it means there is a fundamental limitation for classical electrodynamics (dimension needs to be >1e-25m).

The author seems to believe the infinite result is "correct" within classical electrodynamics and because the result is unacceptable, he suggests the solution is to limit application of classical electrodynamics to bodies that have finite charge density.
This would be correct if by "classical electrodynamics" we mean a theory where we have to use the first integral formula. This formula does not work for point charged particles.
But if by "classical electrodynamics" we mean Maxwell's eqautions and the Lorentz force formula, there is no necessity to use that formula. For point particles at rest, the electrostatic formula with sums is appropriate and works well for point charges. There is also generalization to general situation where particles move and accelerate, so the field is not static. See for example the papers by Frenkel,  Feynman and Wheeler, and Stabler:
J. Frenkel, Zur Elektrodynamik punktfoermiger Elektronen, Zeits. f. Phys., 32, (1925), p. 518-534. http://dx.doi.org/10.1007/BF01331692
J. A. Wheeler, R. P. Feynman, Classical Electrodynamics in Terms of Direct Interparticle Interaction, Rev. Mod. Phys., 21, 3, (1949), p. 425-433. http://dx.doi.org/10.1103/RevModPhys.21.425
R. C. Stabler, A Possible Modification of Classical Electrodynamics, Physics Letters, 8, 3, (1964), p. 185-187. http://dx.doi.org/10.1016/S0031-9163(64)91989-4
