We present a study of 16 planetary nebulae (PNe) where fullerenes have been detected in their Spitzer Space Telescope spectra. This large sample of objects offers a unique opportunity to test conditions of fullerene formation and survival under different metallicity environments because we are analyzing five sources in our own Galaxy, four in the Large Magellanic Cloud (LMC), and seven in the Small Magellanic Cloud (SMC). Among the 16 PNe studied, we present the first detection of C60 (and possibly also C70) fullerenes in the PN M 1–60 as well as of the unusual ~6.6, 9.8, and 20 μm features (attributed to possible planar C24) in the PN K 3–54. Although selection effects in the original samples of PNe observed with Spitzer may play a potentially significant role in the statistics, we find that the detection rate of fullerenes in C-rich PNe increases with decreasing metallicity (~5% in the Galaxy, ~20% in the LMC, and ~44% in the SMC) and we interpret this as a possible consequence of the limited dust processing occurring in Magellanic Cloud (MC) PNe. CLOUDY photoionization modeling matches the observed IR fluxes with central stars that display a rather narrow range in effective temperature (~30,000-45,000 K), suggesting a common evolutionary status of the objects and similar fullerene formation conditions. Furthermore, the data suggest that fullerene PNe likely evolve from low-mass progenitors and are usually of low excitation. We do not find a metallicity dependence on the estimated fullerene abundances. The observed C60 intensity ratios in the Galactic sources confirm our previous finding in the MCs that the fullerene emission is not excited by the UV radiation from the central star. CLOUDY models also show that line- and wind-blanketed model atmospheres can explain many of the observed [Ne III]/[Ne II] ratios using photoionization, suggesting that possibly the UV radiation from the central star, and not shocks, is triggering the decomposition of the circumstellar dust grains. With the data at hand, we suggest that the most likely explanation for the formation of fullerenes and graphene precursors in PNe is that these molecular species are built from the photochemical processing of a carbonaceous compound with a mixture of aromatic and aliphatic structures similar to that of hydrogenated amorphous carbon dust.