  The six-fermion production processes $e^+e^-\to q\bar q l^+ l^-\nu\bar\nu$,
with all the lepton flavours and $q=u,d,c,s$, relevant to the study of the
intermediate-mass Higgs boson at future $e^+e^-$ linear colliders, are
analysed. A Monte Carlo program, taking into account the whole set of
tree-level scattering amplitudes and the relevant radiative effects, is
developed to provide integrated cross sections and generation of unweighted
events. The complete calculation is compared with the available results of real
Higgs production, and the opportunities of precision studies with event
generation are discussed, demonstrating the relevance of a full six-fermion
calculation. Numerical results for integrated cross sections with various
kinematical cuts and including radiative effects are given and commented. In
the analysis of event samples, several distributions are studied and found to
be sensitive to the presence and to the properties of the Higgs boson.
