We present new UV-to-IR stellar photometry of four low-extinction windows in the Galactic bulge, obtained with the Wide Field Camera 3 on the Hubble Space Telescope (HST). Using our five bandpasses, we have defined reddening-free photometric indices sensitive to stellar effective temperature and metallicity. We find that the bulge populations resemble those formed via classical dissipative collapse: each field is dominated by an old (~10 Gyr) population exhibiting a wide metallicity range (-1.5 < [Fe/H] < 0.5). We detect a metallicity gradient in the bulge population, with the fraction of stars at super-solar metallicities dropping from 41% to 35% over distances from the Galactic center ranging from 0.3 to 1.2 kpc. One field includes candidate exoplanet hosts discovered in the SWEEPS HST transit survey. Our measurements for 11 of these hosts demonstrate that exoplanets in the distinct bulge environment are preferentially found around high-metallicity stars, as in the solar neighborhood, supporting the view that planets form more readily in metal-rich environments.