First, we describe the construction of a new type of gravity-matter models based on the formalism of non-Riemannian space-time volume forms - alternative generally covariant integration measure densities (volume elements) defined in terms of auxiliary antisymmetric tensor gauge fields. Here gravity couples in a non-conventional way to two distinct scalar fields providing a unified Lagrangian action principle description of: (i) the evolution of both “early” and“late” Universe - by the “inflaton” scalar field; (ii) dark energy and dark matter as a unified manifestation of a single material entity - the “darkon” scalar field. A physically very interesting phenomenon occurs when including in addition interactions with the electro-weak model bosonic sector - we obtain a gravity-assisted dynamical generation of electro-weak spontaneous gauge symmetry breaking in the post-inflationary “late” Universe, while the Higgs-like scalar remains massless in the “early” Universe. Next, we proceed to the Wheeler–DeWitt minisuperspace quantization of the above models. The “darkon” field plays here the role of cosmological“time”. In particular, we show the absence of cosmological space-time singularities.