  In this letter, we elaborate further on a Cosmological "Running-Vacuum" type
model for the Universe, suggested previously by the authors within the context
of a string-inspired effective theory in the presence of a Kalb-Ramond (KR)
gravitational axion field which descends from the antisymmetric tensor of the
massless gravitational string multiplet. In the presence of this field, which
has anomalous CP violating interactions with the gravitons, primordial
gravitational waves induce gravitational anomalies, which in turn are
responsible for the appearance of $H^2$ and $H^4$ contributions to the vacuum
energy density, these terms being characteristic of generic
"running-vacuum-model (RVM) type", where $H$ is the Hubble parameter. In this
work we prove in detail the appearance of the $H^4$ terms due to
gravitational-anomaly-induced condensates in the energy density of the
primordial Universe, which can self-consistently induce inflation, and
subsequent exit from it, according to the generic features of RVM. We also
argue in favour of the robustness of our results, which were derived within an
effective low-energy field theory approach, against Ultra Violet completion of
the theory. During the radiation and matter-dominated eras, gravitational
anomalies cancel, as required for the consistency of the quantum
matter/radiation field theory. However, chiral and QCD-axion-type anomalies
survive and have important consequences for both cosmic magnetogenesis and
axionic dark matter in the Universe. Finally, the stringy RVM scenario
presented here predicts quintessence-like dynamical dark energy for the current
Universe, which is compatible with the existing fitting analyses of such model
against observations
