About 3%–10% of Type I active galactic nuclei (AGNs) have double-peaked broad Balmer lines in their optical spectra originating from the motion of gas in their accretion disk. Double-peaked profiles arise not only in AGNs, but occasionally appear during optical flares from tidal disruption events and changing-state AGNs. In this paper, we identify 250 double-peaked emitters (DPEs) among a parent sample of optically variable broad-line AGNs in the Zwicky Transient Facility (ZTF) survey, corresponding to a DPE fraction of 19%. We model spectra of the broad Hα emission-line regions and provide a catalog of the fitted accretion disk properties for the 250 DPEs. Analysis of power spectra derived from the 5 yr ZTF light curves finds that DPE light curves have similar amplitudes and power-law indices to other broad-line AGNs. Follow-up spectroscopy of 12 DPEs reveals that ∼50% display significant changes in the relative strengths of their red and blue peaks over long 10–20 yr timescales, indicating that broad-line profile changes arising from spiral arm or hotspot rotation are common among optically variable DPEs. Analysis of the accretion disk parameters derived from spectroscopic modeling provides evidence that DPEs are not in a special accretion state, but are simply normal broad-line AGNs viewed under the right conditions for the accretion disk to be easily visible. We include inspiraling supermassive black hole binary candidate SDSSJ1430+2303 in our analysis, and discuss how its photometric and spectroscopic variability is consistent with the disk-emitting AGN population in the ZTF survey.