We report multiepoch 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 and 0.9km/s 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 unresolved binary candidates, the T0+T4.5 2MASS J11061197+2754225 and the L7+T3.5 2MASSJ21265916+7617440, with orbital periods of 4 and 12yr, respectively. We find a kinematic age of 3.5+/-0.3Gyr for local T dwarfs, consistent with nearby late M dwarfs (4.1+/-0.3Gyr). Removal of thick-disk L dwarfs in the local ultracool dwarf sample gives a similar age for L dwarfs (4.2+/-0.3Gyr), 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 ~20pc of the Sun.