  We simulate the coupled stellar and tidal evolution of short-period binary
stars (orbital period $P_{orb} \lsim$8 days) to investigate the orbital
oscillations, instellation cycles, and orbital stability of circumbinary
planets (CBPs). We consider two tidal models and show that both predict an
outward-then-inward evolution of the binary's semi-major axis $a_{bin}$ and
eccentricity $e_{bin}$. This orbital evolution drives a similar evolution of
the minimum CBP semi-major axis for orbital stability. By expanding on previous
models to include the evolution of the mass concentration, we show that the
maximum in the CBP orbital stability limit tends to occur 100 Myr after the
planets form, a factor of 100 longer than previous investigations. This result
provides further support for the hypothesis that the early stellar-tidal
evolution of binary stars has removed CBPs from short-period binaries. We then
apply the models to Kepler-47 b, a CBP orbiting close to its host stars'
stability limit, to show that if the binary's initial $e_{bin} \gsim$0.24, the
planet would have been orbiting within the instability zone in the past and
probably wouldn't have survived. For stable, hypothetical cases in which the
stability limit does not reach a planet's orbit, we find that the amplitudes of
$a_{bin}$ and $e_{bin}$ oscillations can damp by up to 10\% and 50\%,
respectively. Finally, we consider equal-mass stars with $P_{orb} =$ 7.5 days
and compare the HZ to the stability limit. We find that for stellar masses
$\lsim0.12M_{\odot}$, the HZ is completely unstable, even if the binary orbit
is circular. For $e_{bin} \lsim$0.5, that limit increases to $0.17M_{\odot}$,
and the HZ is partially destabilized for stellar masses up to $0.45M_{\odot}$.
These results may help guide searches for potentially habitable CBPs, as well
as characterize their evolution and likelihood to support life after they are
found.
