  Context. Kepler-444 is one of the oldest planetary systems known thus far.
Its peculiar configuration consisting of five sub-Earth-sized planets orbiting
the companion to a binary stellar system makes its early history puzzling.
Moreover, observations of HI-Ly-$\rm \alpha$ variations raise many questions
about the potential presence of escaping atmospheres today. Aims. We aim to
study the orbital evolution of Kepler-444-d and Kepler-444-e and the impact of
atmospheric evaporation on Kepler-444-e. Methods. Rotating stellar models of
Kepler-444-A were computed with the Geneva stellar evolution code and coupled
to an orbital evolution code, accounting for the effects of dynamical,
equilibrium tides and atmospheric evaporation. The impacts of multiple stellar
rotational histories and extreme ultraviolet (XUV) luminosity evolutionary
tracks are explored. Results. Using detailed rotating stellar models able to
reproduce the rotation rate of Kepler-444-A, we find that its observed rotation
rate is perfectly in line with what is expected for this old K0-type star,
indicating that there is no reason for it to be exceptionally active as would
be required to explain the observed HI-Ly-$\rm \alpha$ variations from a
stellar origin. We show that given the low planetary mass ($\sim$ 0.03 M$_{\rm
\oplus}$) and relatively large orbital distance ($\sim$ 0.06 AU) of
Kepler-444-d and e, dynamical tides negligibly affect their orbits, regardless
of the stellar rotational history considered. We point out instead how
remarkable the impact is of the stellar rotational history on the estimation of
the lifetime mass loss for Kepler-444-e. We show that, even in the case of an
extremely slow rotating star, it seems unlikely that such a planet could retain
a fraction of the initial water-ice content if we assume that it formed with a
Ganymede-like composition.
