What would the electromagnetic field of a massless electron look like The Standard Model gives non-zero mass to the electron via the coupling to the Higgs field. Issues of renormalizability aside, this is fundamentally unrelated to the fact that the electron couples to the EM field. However, if the Higgs mechanism did not operate - that is, if there were no vacuum symmetry breaking, the electron field would have no effective mass term. In QFT perturbation theory, this model offers no special difficulty.
My question is, what is the classical limit of this theory, if it has one? Does the electron acquire a purely EM mass? If the low energy renormalized mass is set to zero (is there an obstacle to doing that?) what do the classical field configurations look like? My puzzlement is related to the classical description of the EM field of a relativistic charged particle - namely, that it becomes "flattened" in the direction of motion, just like a Lorentz-contracted sphere. That is, the field is weaker than for a motionless particle in the longitudinal direction, and stronger in the transverse directions. The limiting case is that the field becomes concentrated on a 2-d surface transverse to the particle location, where it has infinite strength - obviously an unphysical situation. So what actually happens?
 A: This has some bearing on the question in your title
http://arXiv.org/abs/hep-th/9112020
There is an analogous problem in general relativity, the gravitational field generated by a massless particle is called the Aichelburg - Sexl metric. 
Couple more comments:
Perturbation theory in the presence of massless charged fields is not unproblematic, there are infrared divergences one has to deal with. These are much more problematic than soft photon divergences. I think Weinberg's field theory book has a discussion of this.
Massless electrons would have an extra chiral symmetry, which should forbid the appearance of mass term.
But, I don't think these quantum field theory issues are directly related to your question, which is essentially one in classical electrodynamics.
A: With a massless charged fermion, the electromagnetic coupling strength will be totally screened, with $1/e^2$ increasing logarithmically with the scale $R$. The electric field will be Coulomb multiplied by this logarithmic factor in $R$.
Actually, the massless charged fermion will be travelling at the speed of light, but this logarithmic screening will still happen.
