  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.
