  Understanding how supermassive black holes (SMBHs) pair and merge helps to
inform predictions of off-center, dual, and binary AGN, and provides key
insights into how SMBHs grow and co-evolve with their galaxy hosts. As the
loudest known gravitational wave source, binary SMBH mergers also hold
centerstage for the Laser Interferometer Space Antenna (LISA), a joint ESA/NASA
gravitational wave observatory set to launch in 2034. Here, we continue our
work to characterize SMBH binary formation and evolution through increasingly
more realistic high resolution direct $N$-body simulations, focusing on the
effect of SMBH mass ratio, orientation, and eccentricity within a rotating and
flattened stellar host. During the dynamical friction phase, we found a
prolonged orbital decay for retrograde SMBHs and swift pairing timescales for
prograde SMBHs compared to their counterparts in non-rotating models, an effect
that becomes more pronounced for smaller mass ratios $M_{\rm sec}/M_{\rm prim}
= q$. During this pairing phase, the eccentricity dramatically increases for
retrograde configurations, but as the binary forms, the orbital plane flips so
that it is almost perfectly prograde, which stifles the rapid eccentricity
growth. In prograde configurations, SMBH binaries form and remain at
comparatively low eccentricities. As in our prior work, we note that the center
of mass of a prograde SMBH binary itself settles into an orbit about the center
of the galaxy. Since even the initially retrograde binaries flip their orbital
plane, we expect few binaries in rotating systems to reside at rest in the
dynamic center of the host galaxy, though this effect is smaller as $q$
decreases.
