  We investigate several fundamental characteristics of the multiloop
functional renormalization group (mfRG) flow by hands of its application to a
prototypical many-electron system: the Anderson impurity model (AIM). We first
analyze the convergence of the algorithm in the different parameter regions of
the AIM. As no additional approximation is made, the multiloop series for the
local self-energy and response functions converge perfectly to the
corresponding results of the parquet approximation (PA) in the weak- to
intermediate-coupling regime. Small oscillations of the mfRG solution as a
function of the loop order gradually increase with the interaction, hindering a
full convergence to the PA in the strong-coupling regime, where perturbative
resummation schemes are no longer reliable. By exploiting the converged
results, we inspect the fulfillment of (i) sum rules associated to the Pauli
principle and (ii) Ward identities related to conservation laws. For the Pauli
principle, we observe a systematic improvement by increasing the loop order and
including the multiloop corrections to the self-energy. This is consistent with
the preservation of crossing symmetries and two-particle self-consistency in
the PA. For the Ward identities, we numerically confirm a visible improvement
by means of the Katanin substitution. At weak coupling, violations of the Ward
identity are further reduced by increasing the loop order in mfRG. In this
regime, we also determine the precise scaling of the deviations of the Ward
identity as a function of the electronic interaction. For larger interaction
values, the overall behavior becomes more complex, and the benefits of the
higher-loop terms are mostly present in the contributions at large frequencies.
