Do free electrons travel to region of higher potential or lower potential?
The free electrons experiences electrostatic force in a direction opposite to the direction of electric field being is of negative charge. The electric field always directed from higher potential to lower travel. Therefore, electrostatic force and hence direction of travel of electrons is from lower potential to region of higher potential .
Can there be a potential difference between two adjacent conductors carrying the same charge?
Can the potential function have a maximum or minimum in free space?
No, The absence of atmosphere around conductor prevents the phenomenon of electric discharge or potential leakage and hence, potential function do not have a maximum or minimum in free space.
A test charge $q$ is made to move in the electric field of a point charge $Q$ along two different closed paths [figure first path has sections along and perpendicular to lines of electric field. Second path is a rectangular loop of the same area as the first loop. How does the work done compare in the two cases?
As electric field is conservative, work done will be zero in both the cases.
Prove that a closed equipotential surface with no charge within itself must enclose an equipotential volume.
Let's assume contradicting statement that the potential is not same inside the closed equipotential surface. Let the potential just inside the surface is different to that of the surface causing in a potential gradient $\left(\frac{d V}{d r}\right)$. Consequently electric field comes into existence, which is given by as $E=-\frac{d V}{d r}$.
Consequently field lines pointing inwards or outwards from the surface. These lines cannot be again on the surface, as the surface is equipotential. It is possible only when the other end of the field lines are originated from the charges inside.
This contradict the original assumption. Hence, the entire volume inside must be equipotential.