We present ALMA observations of the CO(6-5) and [CII] emission lines and the sub-millimeter continuum of the $z\sim 6$ 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 $2.9\pm 0.5$ kpc, and of $1.4\pm 0.2$ kpc for the 91.5 GHz dust continuum. By assuming that CO(6-5) is thermalized, and by adopting a CO--to--$H2$ conversion factor $\alpha CO=0.8 M\odot K-1 (km/s)-1 pc2$, we infer a molecular gas mass of $M(H2)=(3.2\pm 0.2)\times 1010M\odot$. Assuming that the observed CO velocity gradient is due to an inclined rotating disk, we derive a dynamical mass of $Mdyn sin2(i)=(2.4\pm 0.5)\times 1010 M\odot$, 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: $MBH=(1.8\pm 0.5)\times 109 M\odot$. We find a molecular gas fraction of $\mu =M(H2)/M\ast \sim 4.4$, where $M\ast \approx Mdyn-M(H2)-M(BH)$. We derive a ratio $vrot/\sigma \approx 1-2$ 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.