Let me try to shoot answers here :)
Strange that photons carry energy from one charged particle to another, but not any info about the sign of the charge of the particle.
It does not: it is a virtual photon in fact. Just a way to model what is happening.
So I think the information of the sign of electron charge is carried by the electron field not the electromagnetic field. (Is that correct?)
Exactly.
The non-relativistic Schrodinger equation wave functions are probabilities, whereas the relativistic Dirac equation wavefunctions don't have a clear probabilistic interpretation. (Is this right?)
This is more the problem for the solution of the Klein-Gordon equation than the DIrac one In this last case, one can construct a probability density. But in fact, in all cases, it is sufficient to multiply it with the electric charge of the field, and we end up with a charge distribution that does not have to be positively defined.
There are also problems of fitting pair production and anihilation into both equations. (Do you agree?)
I don't know, what do you mean? :D