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We present new, deep, narrow- and broad-band Hubble Space Telescope observations of seven of the most star-forming brightest cluster galaxies (BCGs). Continuum-subtracted [O II] maps reveal the detailed, complex structure of warm ($T \sim 10^4$ K) ionized gas filaments in these BCGs, allowing us to measure spatially-resolved star formation rates (SFRs) of ~60-600 Msun/yr. We compare the SFRs in these systems and others from the literature to their intracluster medium (ICM) cooling rates (dM/dt), measured from archival Chandra X-ray data, finding a best-fit relation of log(SFR) = (1.67+/-0.17) log(dM/dt) + (-3.25+/-0.38) with an intrinsic scatter of 0.39+/-0.09 dex. This steeper-than-unity slope implies an increasingly efficient conversion of hot ($T \sim 10^7$ K) gas into young stars with increasing dM/dt, or conversely a gradual decrease in the effectiveness of AGN feedback in the strongest cool cores. We also seek to understand the physical extent of these multiphase filaments that we observe in cluster cores. We show, for the first time, that the average extent of the multiphase gas is always smaller than the radii at which the cooling time reaches 1 Gyr, the tcool/tff profile flattens, and that X-ray cavities are observed. This implies a close connection between the multiphase filaments, the thermodynamics of the cooling core, and the dynamics of X-ray bubbles. Interestingly, we find a one-to-one correlation between the average extent of cool multiphase filaments and the radius at which the cooling time reaches 0.5 Gyr, which may be indicative of a universal condensation timescale in cluster cores.