  [ABRIDGED] In this study, we set out to a) demonstrate the sensitivity to <4
R_E transiting planets with periods of a few days around our program stars, and
b) improve our knowledge of some astrophysical properties(e.g., activity,
rotation) of our targets by combining spectroscopic information and our
differential photometric measurements. We achieve a typical nightly RMS
photometric precision of ~5 mmag, with little or no dependence on the
instrumentation used or on the details of the adopted methods for differential
photometry. The presence of correlated (red) noise in our data degrades the
precision by a factor ~1.3 with respect to a pure white noise regime. Based on
a detailed stellar variability analysis, a) we detected no transit-like events;
b) we determined photometric rotation periods of ~0.47 days and ~0.22 days for
LHS 3445 and GJ 1167A, respectively; c) these values agree with the large
projected rotational velocities (~25 km/s and ~33 km/s, respectively) inferred
for both stars based on the analysis of archival spectra; d) the estimated
inclinations of the stellar rotation axes for LHS 3445 and GJ 1167A are
consistent with those derived using a simple spot model; e) short-term,
low-amplitude flaring events were recorded for LHS 3445 and LHS 2686. Finally,
based on simulations of transit signals of given period and amplitude injected
in the actual (nightly reduced) photometric data for our sample, we derive a
relationship between transit detection probability and phase coverage. We find
that, using the BLS search algorithm, even when phase coverage approaches 100%,
there is a limit to the detection probability of ~90%. Around program stars
with phase coverage >50% we would have had >80% chances of detecting planets
with P<1 day inducing fractional transit depths >0.5%, corresponding to minimum
detectable radii in the range 1.0-2.2 R_E. [ABRIDGED]
