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I review a (3+1)-dimensional, string-inspired cosmological model with gravitational anomalies (of Chern-Simons (CS) type) at early epochs, and a totally-antisymmetric torsion, dual to a massless axion-like field ("gravitational axion"), which couples to the CS term. Under appropriate conditions, primordial gravitational waves can condense, leading to a condensate of the CS anomaly term. As a consequence, one obtains inflation in this theory, of running-vacuum-model (RVM) type, without the need for external inflatons. At the end of the inflationary era, chiral fermionic matter is generated, whose gravitational anomalies cancel the primordial ones. On the other hand, chiral anomalies of gauge type, which are also generated by the chiral matter, remain present during the post-inflationary epochs and become responsible for the generation of a non-perturbative mass for the torsion-related gravitational axion, which, in this way, might play the role of a Dark Matter component of geometrical origin. Moreover, in this model, stringy non-perturbative effects during the RVM inflationary phase generate periodic structures for the potential of axion-like particles that arise due to compactification, and co-exist with the gravitational axions. Such periodic potential modulations may lead to an enhanced production of primordial black holes during inflation, which in turn affects the profile of the generated gravitational waves during the radiation era, with potentially observable consequences. This model also entails an unconventional mechanism for Leptogenesis, due to Lorentz-violating backgrounds of the gravitational axions that are generated during inflation, as a consequence of the anomaly condensates, and remain undiluted in the radiation era.