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In this letter, we elaborate further on a Cosmological "Running-Vacuum" typemodel for the Universe, suggested previously by the authors within the contextof a string-inspired effective theory in the presence of a Kalb-Ramond (KR)gravitational axion field which descends from the antisymmetric tensor of themassless gravitational string multiplet. In the presence of this field, whichhas anomalous CP violating interactions with the gravitons, primordialgravitational waves induce gravitational anomalies, which in turn areresponsible for the appearance of $H^2$ and $H^4$ contributions to the vacuumenergy density, these terms being characteristic of generic"running-vacuum-model (RVM) type", where $H$ is the Hubble parameter. In thiswork we prove in detail the appearance of the $H^4$ terms due togravitational-anomaly-induced condensates in the energy density of theprimordial Universe, which can self-consistently induce inflation, andsubsequent exit from it, according to the generic features of RVM. We alsoargue in favour of the robustness of our results, which were derived within aneffective low-energy field theory approach, against Ultra Violet completion ofthe theory. During the radiation and matter-dominated eras, gravitationalanomalies cancel, as required for the consistency of the quantummatter/radiation field theory. However, chiral and QCD-axion-type anomaliessurvive and have important consequences for both cosmic magnetogenesis andaxionic dark matter in the Universe. Finally, the stringy RVM scenariopresented here predicts quintessence-like dynamical dark energy for the currentUniverse, which is compatible with the existing fitting analyses of such modelagainst observations

Quantum Anomalies in String-Inspired Running Vacuum Universe: Inflation and Axion Dark Matter