  We consider - in uniformly strictly convex potential regime - two versions of
random gradient models with disorder. In model (A) the interface feels a bulk
term of random fields while in model (B) the disorder enters though the
potential acting on the gradients. We assume a general distribution on the
disorder with uniformly-bounded finite second moments.
  It is well known that for gradient models without disorder there are no Gibbs
measures in infinite-volume in dimension $d = 2$, while there are
shift-invariant gradient Gibbs measures describing an infinite-volume
distribution for the gradients of the field, as was shown by Funaki and Spohn.
Van Enter and Kuelske proved in 2008 that adding a disorder term as in model
(A) prohibits the existence of such gradient Gibbs measures for general
interaction potentials in $d = 2$. In Cotar and Kuelske (2012) we proved the
existence of shift-covariant random gradient Gibbs measures for model (A) when
$d\geq 3$, the disorder is i.i.d and has mean zero, and for model (B) when
$d\geq 1$ and the disorder has stationary distribution.
  In the present paper, we prove existence and uniqueness of shift-covariant
random gradient Gibbs measures with a given expected tilt $u\in R^d$ and with
the corresponding annealed measure being ergodic: for model (A) when $d\geq 3$
and the disordered random fields are i.i.d. and symmetrically-distributed, and
for model (B) when $d\geq 1$ and for any stationary disorder dependence
structure. We also compute for both models for any gradient Gibbs measure
constructed as in Cotar and Kuelske (2012), when the disorder is i.i.d. and its
distribution satisfies a Poincar\'e inequality assumption, the optimal decay of
covariances with respect to the averaged-over-the-disorder gradient Gibbs
measure.
