  Spurred by the discovery of a boson resonance at the LHC as the result of the
search for the Standard Model Higgs, we pursue our investigation of the
properties and signatures of Higgses in an effective supersymmetric scenario
that goes beyond the usual MSSM. Such scenarios were first introduced to
alleviate the naturalness problem of the MSSM Higgs and are found to have a
very rich phenomenology that allows departures from the Standard Model in the
production rate of the Higgs in many of the search channels. We now include the
constraints from flavour observables in particular the rare decays b-> s gamma
and Bs -> mu+ mu- including the recent measurement from LHCb. We also address
the issue of Dark Matter and its impact on Higgs physics. In particular, we
incorporate the latest data from XENON100 on the spin independent direct
detection rates. These turn out to be powerful constraints, especially if one
also imposes that the observed thermal relic density is obtained. We also study
models with a low abundance that can more easily evade the direct detection
rates. We study the impact of the flavour and Dark Matter observables on the
production rates of the Higgs at the LHC, and their correlations in the
diphoton, diphoton+jets and 4 leptons. We also comment on the other channels.
