  Atomic Fermi gases have been an ideal platform for simulating conventional
and engineering exotic physical systems owing to their multiple tunable control
parameters. Here we investigate the effects of mixed dimensionality on the
superfluid and pairing phenomena of a two-component ultracold atomic Fermi gas
with a short-range pairing interaction, while one component is confined on a
one-dimensional (1D) optical lattice whereas the other is in a homogeneous 3D
continuum. We study the phase diagram and the pseudogap phenomena throughout
the entire BCS-BEC crossover, using a pairing fluctuation theory. We find that
the effective dimensionality of the non-interacting lattice component can
evolve from quasi-3D to quasi-1D, leading to strong Fermi surface mismatch.
Upon pairing, the system becomes effectively quasi-two dimensional in the BEC
regime. The behavior of $T_c$ bears similarity to that of a regular 3D
population imbalanced Fermi gas, but with a more drastic departure from the
regular 3D balanced case, featuring both intermediate temperature superfluidity
and possible pair density wave ground state. Unlike a simple 1D optical lattice
case, $T_c$ in the mixed dimensions has a constant BEC asymptote.
