We report multi-epoch radial velocities, rotational velocities, and atmospheric parameters for 37 T-type brown dwarfs observed with Keck/NIRSPEC. Using a Markov Chain Monte Carlo forward-modeling method, we achieve median precisions of 0.5 km s$^{-1}$ and 0.9 km s$^{-1}$ for radial and rotational velocities, respectively. All of the T dwarfs in our sample are thin disk brown dwarfs. We confirm previously reported moving group associations for four T dwarfs. However, the lack of spectral indicators of youth in two of these sources suggests that these are chance alignments. We confirm two previously un-resolved binary candidates, the T0+T4.5 2MASS J11061197+2754225 and the L7+T3.5 2MASS J21265916+7617440, with orbital periods of 4 yr and 12 yr, respectively. We find a kinematic age of 3.5$pm$0.3 Gyr for local T dwarfs, consistent with nearby late-M dwarfs (4.1$pm$0.3 Gyr). Removal of thick disk L dwarfs in the local ultracool dwarf sample gives a similar age for L dwarfs (4.2$pm$0.3 Gyr), largely resolving the local L dwarf age anomaly. The kinematic ages of local late-M, L, and T dwarfs can be accurately reproduced with population simulations incorporating standard assumptions of the mass function, star formation rate, and brown dwarf evolutionary models. A kinematic dispersion break is found at the L4$-$L6 subtypes, likely reflecting the terminus of the stellar Main Sequence. We provide a compilation of precise radial velocities for 172 late-M, L, and T dwarfs within $sim$20 pc of the Sun.
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