(N/A) The magnetic force on a charged particle is given by $\vec{F} = q(\vec{v} \times \vec{B})$. For the particle to travel undeflected,the magnetic force must be zero. This happens if the velocity $\vec{v}$ is parallel or anti-parallel to the magnetic field $\vec{B}$. Thus,the initial velocity of the particle is either parallel or anti-parallel to the magnetic field.
$(b)$ Yes,the final speed of the charged particle will be equal to its initial speed. The magnetic force acts perpendicular to the velocity vector at all times,meaning it does no work on the particle $(W = \vec{F} \cdot \vec{d} = 0)$. According to the work-energy theorem,the kinetic energy and hence the speed remains constant.
$(c)$ The electron travels from West to East. The electrostatic field is from North to South,so the electric force on the electron (which is negative) acts from South to North. To keep the electron undeflected,the magnetic force must act from North to South. Using Fleming's left-hand rule for a negative charge,the magnetic field must be applied in a vertically downward direction.