  We present ALMA observations of the CO(6-5) and [CII] emission lines and the
sub-millimeter continuum of the $z\sim6$ quasi-stellar object (QSO) SDSS
J231038.88+185519.7. Compared to previous studies, we have analyzed a synthetic
beam that is ten times smaller in angular size, we have achieved ten times
better sensitivity in the CO(6-5) line, and two and half times better
sensitivity in the [CII] line, enabling us to resolve the molecular gas
emission. We obtain a size of the dense molecular gas of $2.9\pm0.5$ kpc, and
of $1.4\pm0.2$ kpc for the 91.5 GHz dust continuum. By assuming that CO(6-5) is
thermalized, and by adopting a CO--to--$H_2$ conversion factor $\rm \alpha_{CO}
= 0.8~ M_{\odot}~K^{-1}~ (km/s)^{-1} ~pc^{2}$, we infer a molecular gas mass of
$\rm M(H_2)=(3.2 \pm0.2) \times 10^{10}\rm M_{\odot}$. Assuming that the
observed CO velocity gradient is due to an inclined rotating disk, we derive a
dynamical mass of $\rm M_{dyn}~sin^2(i) = (2.4\pm0.5) \times 10^{10}~
M_{\odot}$, which is a factor of approximately two smaller than the previously
reported estimate based on [CII]. Regarding the central black hole, we provide
a new estimate of the black hole mass based on the C~IV emission line detected
in the X-SHOOTER/VLT spectrum: $\rm M_{BH}=(1.8\pm 0.5) \times 10^{9}~
M_{\odot}$. We find a molecular gas fraction of $\rm \mu=M(H_2)/M^*\sim4.4$,
where $\rm M^*\approx M_{dyn} - M(H_2)-M(BH)$. We derive a ratio
$v_{rot}/\sigma \approx 1-2$ suggesting high gas turbulence, outflows/inflows
and/or complex kinematics due to a merger event. We estimate a global Toomre
parameter $Q\sim 0.2-0.5$, indicating likely cloud fragmentation. We compare,
at the same angular resolution, the CO(6-5) and [CII] distributions, finding
that dense molecular gas is more centrally concentrated with respect to [CII].
We find that the current BH growth rate is similar to that of its host galaxy.
