  M dwarf stars are exciting targets for exoplanet investigations; however,
their fundamental stellar properties are difficult to measure. Perhaps the most
challenging property to measure is stellar age. Once on the main sequence, M
dwarfs change imperceptibly in their temperature and luminosity, necessitating
novel statistical techniques for estimating their ages. In this paper, we infer
ages for known eccentric-planet-hosting M dwarfs using a combination of
kinematics and $\alpha$-element-enrichment, both shown to correlate with age
for Sun-like FGK stars. We calibrate our method on FGK stars in a Bayesian
context. To measure $\alpha$-enrichment, we use publicly-available spectra from
the CARMENES exoplanet survey and a recently developed [Ti/Fe] calibration
utilizing individual Ti I and Fe I absorption lines in $Y$ band. Tidal effects
are expected to circularize the orbits of short-period planets on short
timescales; however, we find a number of mildly eccentric, close-in planets
orbiting old ($\sim$8 Gyr) stars. For these systems, we use our ages to
constrain the tidal dissipation parameter of the planets, $Q_\mathrm{p}$. For
two mini-Neptune planets, GJ 176b and GJ 536b, we find they have $Q_\mathrm{p}$
values more similar to the ice giants than the terrestrial planets in our Solar
System. For GJ 436b, we estimate an age of $8.9^{+2.3}_{-2.1}$ Gyr and
constrain the $Q_\mathrm{p}$ to be $>10^5$, in good agreement with constraints
from its inferred tidal heating. We find that GJ 876d has likely undergone
significant orbital evolution over its $8.4^{+2.2}_{-2.0}$ Gyr lifetime,
potentially influenced by its three outer companions which orbit in a Laplace
resonance.
