  We propose a simple experimental scheme to realize simultaneously the
one-dimensional spin-orbit coupling and the staggered spin-flip in ultracold
pseudospin-$1/2$ atomic Fermi gases trapped in square optical lattices. In the
absence of interspecies interactions, the system supports gapped Chern
insulators and gapless topological semimetal states. By turning on the $s$-wave
interactions, a rich variety of gapped and gapless inhomogeneous topological
superfluids can emerge. In particular, a gapped topological Fulde-Ferrell
superfluid, in which the chiral edge states at opposite boundaries possess the
same chirality, is predicted.
