Electrons have their own individual electric field. Currents exist because the electrons are responding to an external field. It is the external field, not the electric field belonging to the electrons, that is responsible for the flow of charge. Everything that responds to an external electric field has its own.
Electrons - or any particles, for that matter - do not have their own magnetic field. Magnetic fields don't work that way. What we consider a 'magnetic field' - that responsible for the Lorentz force - is a consequence of moving charges. Magnetic fields are what happens when you observe an electric field in a relativistic reference frame - from the point of view of the moving charge, the electric fields have a different geometry, and so an additional force is present.
Your suggestion that the fields are 'fixed in place' is, I suppose, true, but I wouldn't feel comfortable describing them that way. There's nothing 'fixing' a magnetic field in place; being constant in space is what a magnetic field is, at least when you have constant current. In other words, saying that the field itself could move is nonsensical; fields don't 'move'; they change in space. If you have a non-constant current, or a current-carrying object moving in space, then yes in a particular fixed position the stength and orientation of the magnetic field will change. But a magnetic field is not bound to any particular object or charge, it is what it is as a consequence of how it is generated.