doubts regarding the classical electron radius When estimating the classical electron radius, people normally equate the energy needed to assemble a charged sphere to $mc^2$, due to the so-called mass-energy relation. However, personally I don't find this very convincing. Imagining we have another sphere with the same size and same mass that is uncharged, then the difference between the classical electron and this sphere is that the latter does not have electrostatic potential energy, but still it has a rest energy of $mc^2$.
My point is when it comes to the energy of an electron at rest, I think the electrostatic energy and the rest energy should be additive instead of being equated with each other. So what's the problem here?
 A: "Assembling" is a hand-wave explanation, it is not serious. In CED the electron is point-like, but in some calculations (scattering or something) a dimensional quantity $r_0$ arises. It does not mean the electron size, though.
A: The classical electron radius has no basis in fact.
And potential energy is associated with a system not with any specific particle, is frame dependent, sometime gauge dependent, and non local son relativistically problematic. It is not a store of energy.
In the case of electromagnetism, energy is stored in volumes containing electromagnetic fields (and in certain surfaces, including surfaces at infinity if space is infinite). So energy is actually transferred between charges and fields when electromagnetic fields exert forces on charged particles.
Given that, you might consider the energy stored in the field of a small spherically symmetric electric field. You'll find that at some critical radius there is more energy in the fields than the total energy of the particle.
But since electrons have a magnetic moment, they have a magnetic field too, which has energy as well, so you already went too far. So even if your charge was stored on a surface that had no energy associated with the tension holding it together or the charged parts making up the surface then you already have too much energy in the electric field and he magnetic field. So you need to be larger than the classical radius, or have negative energy.
No one thinks the electron is made of energy free charge on a surface of a sphere of radius equal to the classical radius with all the mass of the charge associated with the energy of the electric field. Absolutely no one.
If you assembled an uncharged sphere it wouldn't have to have tension to hold itself apart. And it could have any mass you want because the electron surface was having no energy associated with the actual charge, zero energy at each place where the charge was. So an identical sphere but without the charge would be tension without any energy anywhere and you wouldn't need the tension because there is no charge.
A: Putting aside the electron case, which is covered by Vladimir's answer, the energy of a charged sphere at rest is indeed what you are saying it to be. 
Now, is the uncharged sphere shell mass incomplete? No, it is just not included because is not necessary. 
In Physics, Energy needs to be accounted completely only if is involved actively in the processes occurring. For example, before we ever saw processes that involved changes on the $mc^2$ value, we only needed to account for the types of energies involved actively in processes. So you would only use the potential energies involved, partly because we didn't know, but we didn't know because it was irrelevant for the processes.
So in your case, if you are analyzing the process of assembling a charged sphere, electrostatic potential interaction between its parts is relevant, but shell-mass energy is not involved, and therefore considering it will not change your conclusion since it will be a portion of energy initially which remains in the end, and takes no part in the involved energy.
Now think of this, you have many little spheres, like in a gas, and only one of them is charged, and there are no magnetic nor electric  external sources present. Will this gas differ from an identical one where the charged sphere is replaced by  an uncharged one? Nope, so there you see the irrelevance of a non involved energy.
