1) A charge 5 Coulomb is placed at a point. What is the work required to carry 1 C of charge once round it in a circle of 12 cm radius?

2) Is electrostatic potential neceessarily zero at a point where electric field strenght is zero?

1>When we place a point charge at center of circle and revolve other charge along the circumference of the circle the point charge makes an equipotential surface along the circumference of circle  because it has same distance from centre(radius)
hence the work done to carry 1 coloumb charge around 5 coloumd charge is zero.
2> No, it is not necessary that potential at a point must be zero but the change in potential must be zero
for example take 2 positive charge of 1C each and place at some distance  to each other
then find electric field at mid point you get zero , but the potential due to both charge is not zero they are additive in nature.