  We demonstrate that the generators for the local, continuous and
infinitesimal classical gauge symmetry transformations in the cases of (i) the
St$\ddot u$ckelberg-modified massive Abelian 1-form and 2-form theories, and
(ii) the massless Abelian 1-form and 2-form free theories owe their origin to
the first-class constraints of the these theories. We establish a connection
between the standard forms of the generators and the Noether conserved charges
for the modified massive and massless versions of the above theories. We
discuss the appearance of these constraints, within the framework of
Becchi-Rouet-Stora-Tyutin (BRST) formalism, through the physicality criteria
w.r.t. the conserved and nilpotent (anti-)BRST charges. One of the highlights
of our present investigation is the observation that, in the context of the
modified massive and massless Abelian 2-form theories, the modified forms of
the standard Noether (anti-)BRST charges are required which are found to be
off-shell nilpotent and they lead to the appearance of the operator forms of
the first-class constraints through the physicality criteria at the quantum
level. We also comment on (i) the existence of the Curci-Ferrari (CF)-type
restrictions on the Abelian 2-form theories (with and without mass), (ii) the
modifications in the St$\ddot u$ckelberg-technique for the massive 2D Abelian
1-form and 4D Abelian 2-form theories and their consequences, and (iii) the
off-shell nilpotent version of the conserved co-BRST charge and its role in the
physicality criteria for the St$\ddot u$ckelberg-modified 4D massive Abelian
2-form theory.
