We analyze the deuteron charge and quadrupole form factors using the latest NN potentials and charge-density operators derived within chiral effective field theory. We derive relations between 1N,2N charge densities and nucleon form factors and employ several recent empirical parametrizations for proton and neutron form factors. We fit low-energy constants appearing in the fifth-order 2N charge-density operators to the world data on the shape of deuteron form factors and predict the deuteron structure radius and its quadrupole moment. Using predicted structure radius and precise data on the deuteron-proton charge-radii-difference we extract the charge radius of the neutron. A careful analysis of various sources of the uncertainties of all predictions is performed.