Answer
a) No.
b) Yes.
Work Step by Step
$$\color{blue}{\bf [a]}$$
We know that
$$\Delta V=V_f-V_i=-\int_i^fEds$$
So if $E=0$, then we can conclude that $\Delta V=0$ V, NOT $V=0$ V.
So the answer is $\bf No$, we can not.
$$\color{blue}{\bf [b]}$$
We also know that
$$E_s=-\dfrac{dV}{ds}$$
So if $V=0$ throughout some region which means that $dV=0$, then the electric field must be zero. So we can conclude that $E_s=0$ throughout this region.
The answer is $\bf Yes$, we can.
