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.