work done to move a charge along a closed path inside an electric path inside an electric field is always zero using this fact prove that it is impossible to produce an electric field in which all field lines would be parallel and density of their distirbution would constantly increase in a direction perpendicular to the field line

Yes the work done in moving a change in closed path inside an electric field is zero because electric field is a conservative field. 

It is clear from the statement that, when the density of the distribution is constantly increasing in a direction perpendicular to the field line then in this case field line become more dens ( more close to each other ) as we move perpendicular to the field in that direction (direction of increasing field). Thus in this case it is impossible to produce an electric field lines in which all field lines would be parallel and equidistant.