  The study of subtle effects on transport in semiconductors requires
high-quality epitaxial structures with low defect density. Using hybrid
molecular beam epitaxy (MBE), SrTiO$_3$ films with low-temperature mobility
exceeding 42,000 cm$^2$V$^{-1}$s$^{-1}$ at low carrier density of 3 x 10$^{17}$
cm$^{-3}$ were achieved. A sudden and sharp decrease in residual resistivity
accompanied by an enhancement in the superconducting transition temperature
were observed across the second Lifshitz transition (LT) where the third band
becomes occupied, revealing dominant intra-band scattering. These films further
revealed an anomalous behavior in the Hall carrier density as a consequence of
the antiferrodistortive (AFD) transition and the temperature-dependence of the
Hall scattering factor. Using hybrid MBE growth, phenomenological modeling,
temperature-dependent transport measurements, and scanning superconducting
quantum interference device imaging, we provide critical insights into the
important role of inter- vs intra-band scattering and of AFD domain walls on
normal-state and superconducting properties of SrTiO$_3$.
