  We perform consistent reductions and measurements for thirteen ultra-faint
dwarf galaxies (UFDs) observed with the GIRAFFE spectrograph on the Very Large
Telescope (VLT) between 2009 and 2017. In the first paper of this series, we
locate new members and provide refined measurements of physical parameters for
the Bo{\"o}tes I, Leo IV and Leo V dwarf galaxies. We identify nine new Leo IV
members and four new Leo V members, and perform a comparative analysis of
previously discovered members. Additionally, we identify one new binary star in
both Leo IV and Leo V. After removing binary stars, we recalculate the velocity
dispersions of Bo{\"o}tes I and Leo IV to be 5.1$^{+0.6}_{-0.7}$ and
3.6$^{+1.0}_{-1.1}$ km s$^{-1}$, respectively. We do not resolve the Leo V
velocity dispersion. We identify a weak velocity gradient in Leo V that is
$\sim$4$\times$ smaller than the previously calculated gradient and that has a
corresponding position angle which differs from the literature value by
$\sim$120 deg. Combining the VLT data with previous literature, we re-analyze
the Bo{\"o}tes I metallicity distribution function and find that a model
including infall of pristine gas while Bo{\"o}tes I was forming stars best fits
the data. Our analysis of Leo IV, Leo V and other UFDs will enhance our
understanding of these enigmatic stellar populations and contribute to future
dark matter studies. Similar analyses of the remaining ten UFDs will be
presented in forthcoming papers.
