Electron electric field As we know the fundamental unit of charge in our universe at the time of electrodynamics was an electron, and in any frame of reference, its radius is a finite number and assuming uniform charge distribution inside the electron, I can find the electric field at the origin of the electron by $\nabla \cdot \vec{E}= \rho$ which gives me a finite value of the electric field everywhere and $0$ at the centre of the charge. This is an exact solution for the electron problem.
Why do we bother with treating electrons etc. as point charges and get all the singularities in our electric field, when clearly it doesn't even agree with the theory, let alone experiment.
Is it just an approximation that works well only when you are away from the source and using it for the source itself will give results that aren't true for classical electrodynamics itself ? Are things like line charge ,surface charge also just approximations that will give false and inconsistent results when taken too literally?
and obviously there exists no point charge and no line charge and no surface in our universe, as everything is 3dimensional. So Maxwell's equations don't give discontinuities and infinities but our approximations do.
 A: 
As we know [...] in any frame of reference, its radius is a finite number 

Er ... do we know that? 
Relativity--as we understand it--does not allow for fundamental particles of finite size, and there is no experimental hint at all of any size associated with the electron.
Pointedly the "classical electron radius" is about $2.8 \times 10^{-15} \,\mathrm{m}$, but the electron has been probed for substructure (including non-pointlike charge distribution) down to approximately $10^{-18} \,\mathrm{m}$ and indirectly down to about $10^{-22}\,\mathrm{m}$ without any indication of structure at all. In other words the classical electron radius may be a useful number in some calculation but definitively does not represent the "size" of the electron.

obviously there exists no point charge and no line charge and no surface in our universe, as everything is 3dimensional.

Again, this is anything but obvious. When you attempt to reconcile general relativity with quantum mechanics (preserving the symmetries of known physical laws) you get theories with $N$ spacial dimensions for $N > 3$, and yet somehow have only 3 such dimensions that matter to us on a day to day basis. And these days a lot of people are plying around with the notion that the 3D space we think we live in could be a holographic projection from a 2D horizon.
Fun, izinit?
