  We formulate a self-consistent non-minimal five-parameter
Einstein-Yang-Mills-Higgs (EYMH) model and analyse it in terms of effective
(associated, color and color-acoustic) metrics. We use a formalism of
constitutive tensors in order to reformulate master equations for the gauge,
scalar and gravitational fields and reconstruct in the algebraic manner the
so-called associated metrics for the Yang-Mills field. Using WKB-approximation
we find color metrics for the Yang-Mills field and color-acoustic metric for
the Higgs field in the framework of five-parameter EYMH model. Based on
explicit representation of these effective metrics for the EYMH system with
uniaxial symmetry, we consider cosmological applications for Bianchi-I, FLRW
and de Sitter models. We focus on the analysis of the obtained expressions for
velocities of propagation of longitudinal and transversal color and
color-acoustic waves in a (quasi)vacuum interacting with curvature; we show
that curvature coupling results in time variations of these velocities. We
show, that the effective metrics can be regular or can possess singularities
depending on the choice of the parameters of non-minimal coupling in the
cosmological models under discussion. We consider a physical interpretation of
such singularities in terms of phase velocities of color and color-acoustic
waves, using the terms ``wave stopping'' and ``trapped surface''.
