  We discuss in some detail the properties of gravity (including f(R)-gravity)
coupled to non-standard nonlinear gauge field system containing a square root
of the usual Maxwell Lagrangian. The latter is known to produce in flat
spacetime a QCD-like confinement. Inclusion of gravity triggers various
physically interesting effects: new mechanism for dynamical generation of
cosmological constant; non-standard black hole solutions with constant vacuum
electric field and with "hedge-hog"-type spacetime asymptotics, which are shown
to obey the first law of black hole thermodynamics; new "tubelike" solutions of
Levi-Civita-Bertotti-Robinson type; charge-"hiding" and charge-confining
"thin-shell" wormhole solutions; dynamical effective gauge couplings and
confinement-deconfinement transition effect when coupled to quadratic
R^2-gravity.
