Ultra-faint dwarf galaxies (UFDs) are among the oldest and most metal-poor stellar systems in the Universe. Their metallicity distribution encodes the fossil record of the earliest star formation, feedback, and chemical enrichment, providing crucial tests of models of the first stars, galaxy assembly, and dark matter halos. However, due to their faint luminosities and the limited number of bright giants, spectroscopic studies of UFDs typically probe only small stellar samples. Here, we present an analysis of multi-epoch Hubble Space Telescope and James Webb Space Telescope observations of the UFD Bootes I. Using deep color-magnitude diagram in the F606W and F322W2 bands, extending from the subgiant branch to the M-dwarfs, and stellar proper motions to identify likely members, we obtained an unprecedentedly clean census of the system. The exquisite quality of the diagram, combined with the sensitivity of M-dwarf colors to metallicity, allowed us to constrain the metallicity distribution in a large stellar sample. As a first step, we then exploited the metallicity sensitivity of M-dwarf colors to derive the metallicity distribution function. We find that most of the stars ~85% have [Fe/H]<-2, and that roughly ~17% have [Fe/H]<-3. Then, we derived the binary fraction in Bootes I. This is crucial, since binaries can bias kinematic mass estimates, affect stellar population analyses, and shape the photometric signatures used to infer metallicity. We find that 20+/-2% of stellar systems in Bootes I are binaries with mass ratios larger than 0.4, corresponding to a total binary fraction of ~30%. This value is comparable to the binary fractions observed in globular clusters of similar stellar mass, suggesting that the presence of dark matter does not significantly affect the binary properties of Bootes I.