  We set the stage for reassessing how star formation, emission-line nebulae,
and active galactic nuclei (AGNs) in brightest cluster galaxies (BCGs) depend
on the thermodynamics of the intracluster medium (ICM). Our work is based on
the 25 clusters observed in the CLASH program for which the aforementioned
attributes in their BCGs can be well scrutinized, as has the thermodynamics of
their ICM. Nine of these BCGs display complex UV morphologies tracing recent
star formation, whereas the remaining 16 are characterized by a relatively
compact central UV enhancement. Here, we show definitively that three of the
latter BCGs also display star formation, whereas the diffuse UV of the
remaining 13 is entirely consistent with old low-mass stars. The overall
results support the previously established dependence of star formation and
nebulae in BCGs on an "excess core entropy," K$_{0}$, for the ICM: all 11
clusters with K$_{0}$ $\leq$ 24 keV cm$^{2}$ (but only one of 14 clusters with
K$_{0}$ $\geq$ 42 keV cm$^{2}$) host star-forming BCGs that almost if not
always possess nebulae. Instead of an entropy floor, we show that K$_{0}$
reflects the degree to which the radial entropy profile decreases inward within
$\sim$100 kpc rather than (except perhaps at large K$_{0}$) actually
flattening: clusters with lower ICM entropies and hence shorter cooling times
at their cores preferentially host BCGs displaying star formation, nebulae, and
more radio-luminous AGNs. Nearly all BCGs possess detectable AGNs, however,
indicating multiple pathways for fuelling their AGNs.
