Question

A closed loop of B is produced by a changing electric field. Does it necessarily mean that E and dE/dt are non-zero at all points on the loop and in the area enclosed by the loop?

Answer 1

Not necessarily. All that is needed is that the total electric flux through the area enclosed by the loop should vary in time. The flux change may arise from any portion of the area. Elsewhere E or dE/dt may be zero. In particular, there need be no electric field at the points which make the loop.