  It was recently shown [Phys. Rev. Lett. {\bf 110}, 227201 (2013)] that the
critical behavior of the random-field Ising model in three dimensions is ruled
by a single universality class. This conclusion was reached only after a proper
taming of the large scaling corrections of the model by applying a combined
approach of various techniques, coming from the zero- and positive-temperature
toolboxes of statistical physics. In the present contribution we provide a
detailed description of this combined scheme, explaining in detail the
zero-temperature numerical scheme and developing the generalized
fluctuation-dissipation formula that allowed us to compute connected and
disconnected correlation functions of the model. We discuss the error evolution
of our method and we illustrate the infinite limit-size extrapolation of
several observables within phenomenological renormalization. We present an
extension of the quotients method that allows us to obtain estimates of the
critical exponent $\alpha$ of the specific heat of the model via the scaling of
the bond energy and we discuss the self-averaging properties of the system and
the algorithmic aspects of the maximum-flow algorithm used.
