The answers you seek unfortunately lie in particle physics, quantum electro dynamics which themselves is buried in a lot of complicated maths.
However, in a nutshell: classical physics tells us there is a 'field' that carries the magnetic and electric forces from one body to another. The source of this field is as you said either a moving charge for magnetic or simply a charge for electric. These fundamental properties are like mass, that are inherent properties of the object, ie proton whatever. this can be further broken down into other subatomic particles, namely quarks. each quark has a charge of either plus or minus a third so you combine 3 quarks in different ways to get a 1 or a 0 for proton or neutron. Beyond that, i think its fair to say, and i might be wrong here just because i don't know a whole lot myself, that fundamental charge is something that lies beyond the scope of physics, and possibly human understanding for the moment, as your really delving into philosphy and metaphysics as well, all those 'big' questions.
Particle physics however tells us that fields don't exist, and that all forces are carried by particles. for EM its the photon. when two electrons cross paths and 'interact' with each other, photons are exhcanged in such a way that they deflect. these photons are emited and reabsorbed all the time by electrons along their path, but these are virtual photons. once an itneraction occurs they become real, according to my understanding anyway.
hope this helps
-G