Recent images from the Event Horizon Telescope of accreting supermassive black holes (BHs), along with upcoming observations with better sensitivity and angular resolution, offer exciting opportunities to deepen our understanding of spacetime in strong gravitational fields. A significant focus for future BH imaging observations is the direct detection of the "photon ring," a narrow band on the observer's sky that collects extremely lensed photons. The photon ring consists of self-similarly nested subrings which, in spherically-symmetric spacetimes, are neatly indexed by the maximum number of half-loops executed around the BH by the photons that arrive in them. Each subring represents an entire "higher-order" image of the horizon-scale accretion flow. Furthermore, this self-similarity is controlled by a single critical lensing exponent linked to the radial (in)stability of photon orbits near the critical (circular) photon orbit, solely determined by the spacetime geometry. However, extracting such information about the spacetime geometry can be challenging because the observed photon ring is also influenced by the structure of the emitting region. To address this, we conducted a comprehensive study by varying (a) a wide range of emission-zone morphology models and (b) families of spacetime metrics. We find that the lensing exponent can be reliably determined from future observations. This exponent can provide access to the component of the spacetime metric, as well as significantly narrow down currently accessible BH parameter spaces. Additionally, the width of the first-order photon subring serves as yet another important discriminator of the spacetime geometry. Finally, observations of flaring events across different wavelengths might reveal time-delayed secondary images, with the delay time providing a promising new way to independently estimate the BH shadow size.