  We present a comprehensive analysis of the deuteron charge and quadrupole
form factors based on the latest two-nucleon potentials and charge density
operators derived in chiral effective field theory. The single- and two-nucleon
contributions to the charge density are expressed in terms of the proton and
neutron form factors, for which the most up-to-date empirical parametrizations
are employed. By adjusting the fifth-order short-range terms in the two-nucleon
charge density operator to reproduce the world data on the momentum-transfer
dependence of the deuteron charge and quadrupole form factors, we predict the
values of the structure radius and the quadrupole moment of the deuteron:
$r_{\rm str}=1.9729\substack{+0.0015\\ -0.0012}\ \text{fm},\
Q_d=0.2854\substack{+0.0038\\ -0.0017}\ \text{fm}^2. $ A comprehensive and
systematic analysis of various sources of uncertainty in our predictions is
performed. Following the strategy advocated in our recent publication Phys.
Rev. Lett. 124, 082501 (2020), we employ the extracted structure radius
together with the accurate atomic data for the deuteron-proton mean-square
charge radii difference to update the determination of the neutron charge
radius, for which we find: $r_n^2=-0.105\substack{+0.005\\ -0.006} \,
\text{fm}^2$. Given the observed rapid convergence of the deuteron form factors
in the momentum-transfer range of $Q \simeq 1-2.5$ fm$^{-1}$, we argue that
this intermediate-energy domain is particularly sensitive to the details of the
nucleon form factors and can be used to test different parametrizations.
