  Transfer-matrix methods are used to calculate spin-spin correlation functions
($G$), Helmholtz free energies ($f$) and magnetizations ($m$) in the
two-dimensional random-field Ising model close to the zero-field bulk critical
temperature $T_{c 0}$, on long strips of width $L = 3 - 18$ sites, for binary
field distributions. Analysis of the probability distributions of $G$ for
varying spin-spin distances $R$ shows that describing the decay of their
averaged values by effective correlation lengths is a valid procedure only for
not very large $R$. Connections between field-- and correlation function
distributions at high temperatures are established, yielding approximate
analytical expressions for the latter, which are used for computation of the
corresponding structure factor. It is shown that, for fixed $R/L$, the
fractional widths of correlation-function distributions saturate asymptotically
with $L^{-2.2}$. Considering an added uniform applied field $h$, a connection
between $f(h)$, $m(h)$, the Gibbs free energy $g(m)$ and the distribution
function for the uniform magnetization in zero uniform field, $P_0(m)$, is
derived and first illustrated for pure systems, and then applied for non-zero
random field. From finite-size scaling and crossover arguments, coupled with
numerical data, it is found that the width of $P_0(m)$ varies against
(non-vanishing, but small) random-field intensity $H_0$ as $H_0^{-3/7}$.
