We present the full results of our decade-long astrometric monitoring programs targeting 31 ultracool binaries with component spectral types M7-T5. Joint analysis of resolved imaging from Keck Observatory and Hubble Space Telescope and unresolved astrometry from CFHT/WIRCam yields parallactic distances for all systems, robust orbit determinations for 23 systems, and photocenter orbits for 19 systems. As a result, we measure 38 precise individual masses spanning 30-115 $M_{rm Jup}$. We determine a model-independent substellar boundary that is $approx$70 $M_{rm Jup}$ in mass ($approx$L4 in spectral type), and we validate Baraffe et al. (2015) evolutionary model predictions for the lithium-depletion boundary (60 $M_{rm Jup}$ at field ages). Assuming each binary is coeval, we test models of the substellar mass-luminosity relation and find that in the L/T transition, only the Saumon & Marley (2008) hybrid models accounting for cloud clearing match our data. We derive a precise, mass-calibrated spectral type-effective temperature relation covering 1100-2800 K. Our masses enable a novel direct determination of the age distribution of field brown dwarfs spanning L4-T5 and 30-70 $M_{rm Jup}$. We determine a median age of 1.3 Gyr, and our population synthesis modeling indicates our sample is consistent with a constant star formation history modulated by dynamical heating in the Galactic disk. We discover two triple-brown-dwarf systems, the first with directly measured masses and eccentricities. We examine the eccentricity distribution, carefully considering biases and completeness, and find that low-eccentricity orbits are significantly more common among ultracool binaries than solar-type binaries, possibly indicating the early influence of long-lived dissipative gas disks. Overall, this work represents a major advance in the empirical view of very low-mass stars and brown dwarfs.
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