We conduct precise strong lensing mass modeling of four Hubble Frontier Field (HFF) clusters, Abell 2744, MACS J0416.1-2403, MACS J0717.5+3745, and MACS J1149.6+2223, for which HFF imaging observations are completed. We construct a refined sample of more than 100 multiple images for each cluster by taking advantage of the full-depth HFF images, and conduct mass modeling using the glafic software, which assumes simply parametrized mass distributions. Our mass modeling also exploits a magnification constraint from the lensed SN Ia HFF14Tom for Abell 2744 and positional constraints from the multiple images S1-S4 of the lensed supernova SN Refsdal for MACS J1149.6+2223. We find that our best-fitting mass models reproduce the observed image positions with rms errors of ~0.4", which are smaller than rms errors in previous mass modeling that adopted similar numbers of multiple images. Our model predicts a new image of SN Refsdal with a relative time delay and magnification that are fully consistent with a recent detection of reappearance. We then construct catalogs of z~6-9 dropout galaxies behind the four clusters and estimate magnification factors for these dropout galaxies with our best-fitting mass models. The dropout sample from the four cluster fields contains ~120 galaxies at z>~6, about 20 of which are predicted to be magnified by a factor of more than 10. Some of the high-redshift galaxies detected in the HFF have lensing-corrected magnitudes of M_UV_~-15 to -14. Our analysis demonstrates that the HFF data indeed offer an ideal opportunity to study faint high-redshift galaxies. All lensing maps produced from our mass modeling will be made available on the Space Telescope Science Institute website (https://archive.stsci.edu/prepds/frontier/lensmodels/).