Is a charged particle at rest affected by magnetic field? It is known that particles such as electrons and protons bear electric charge, but not a magnetic charge. When these particles are at rest, are they somehow affected by magnetic field?
The similar question applies to neutrons: are they affected by electric field?
 A: Depends on what how far you are willing to go. If you have a solenoid with a uniform time-varying magnetic field, then an electric field is induced by Faraday's law. It will be a circular field, and will make an electron at rest spiral. So the electron at rest is affected by a time-varying magnetic field, though indirectly--via an induced electric field.
Without considering induced fields:
An electron will align its spin with a uniform magnetic field, and if the field is nonuniform, it will move along with it. This is because the electron has a magnetic "spin" or dipole moment,
As far as neutrons are concerned, it depends if they have an electric dipole moment(afaict unknown). But, they can be affected in a similar manner by time-varying nonuniform electric fields, since they also have a magnetic moment.
A: It is not.  The force on it goes like $F_{\rm Mag} = q\,{\mathbf v} \times {\mathbf B}$ and so if the velocity ${\mathbf v} = {\mathbf 0}$ the force will vanish.
Neutrons have total electric charge which vanishes.  In theory, it could have an electric dipole moment, but this is known experimentally to be quite small and maybe exactly zero.
