Can an electron at rest in a magnetic field be set into motion by the magnetic field? What if it were at rest in an electric field?

Answer 1

No to the first question. The electron has to be moving to experience a force.

Yes to the second question.

There is a paradox here that led to the discovery of the Theory of Relativity. If the observer's frame of reference is moving, a formerly stationary electron will be appear to be moving.

This dilemma was resolved by Lorentz before Einstein came along. When the electron appears to move due to the motion of the observer, some of the electric field changes to magnetic field
with the end result that there is still no net force in either coordinate system.