If the total charge enclosed by a surface is zero, does it imply that the electric field everywhere on the surface is zero ? Conversely, if the electric field everywhere on a surface is zero, does it imply that net charge inside is zero.

Explanation

Gauss' law also implices that when the surface is so chosen that there are some charges inside and some outside.
the flux is such situation is given by `ointE.dS=(q)/(epsi_(0))`
In such situations, the electric field in the LHS is due to all the cahrges both inside and outside the surface. the term q on the right side of the equation given by gauss' law represent only the total charge inside the surface.
Thus, despite being total cahrge enclosed by a surface zero, it doesn't imply that the electric field everywhere on the surface is zero, the field may be normal to the surface. Also, conversely if teh electric field everywhere on a surface is zero, it doesn't imply that net charge inside it is zero.
i.e., putting `E=0` in `ointE.dS=(qw)/(epsi_(0))`
we get `q=0`.