The Galactic center is dominated by the gravity of a super-massive black hole (SMBH), Sagittarius A∗, and is suspected to contain a sizable population of binary stars. Such binaries form hierarchical triples with the SMBH, undergoing Eccentric Kozai-Lidov (EKL) evolution, which can lead to high-eccentricity excitations for the binary companions' mutual orbit. This effect can lead to stellar collisions or Roche-lobe crossings, as well as orbital shrinking due to tidal dissipation. In this work we investigate the dynamical and stellar evolution of such binary systems, especially with regards to the binaries' post-main-sequence evolution. We find that the majority of binaries (∼75%) is eventually separated into single stars, while the remaining binaries (∼25%) undergo phases of common-envelope evolution and/or stellar mergers. These objects can produce a number of different exotic outcomes, including rejuvenated stars, G2-like infrared-excess objects, stripped giant stars, Type Ia supernovae (SNe), cataclysmic variables, symbiotic binaries, or compact object binaries. We estimate that, within a sphere of 250 Mpc radius, about 7.5-15 SNe Ia per year should occur in galactic nuclei due to this mechanism, potentially detectable by the Zwicky Transient Facility and ASAS-SN. Likewise, we estimate that, within a sphere of 1 Gpc3 volume, about 10-20 compact object binaries form per year that could become gravitational wave sources. Based on results of EKL-driven compact object binary mergers in galactic nuclei by Hoang et al., this compact object binary formation rate translates to about 15-30 events per year that are detectable by Advanced LIGO.