  We present a multi-line survey of the interstellar medium (ISM) in two $z>6$
quasar (QSO) host galaxies, PJ231-20 ($z=6.59$) and PJ308-21 ($z=6.23$), and
their two companion galaxies. Observations were carried out using the Atacama
Large (sub-)Millimeter Array (ALMA). We targeted eleven transitions including
atomic fine structure lines (FSLs) and molecular lines: [NII]$_{\rm 205\mu m}$,
[CI]$_{\rm 369\mu m}$, CO ($J_{\rm up} = 7, 10, 15, 16$), H$_2$O
$3_{12}-2_{21}$, $3_{21}-3_{12}$, $3_{03}-2_{12}$, and the OH$_{\rm 163\mu m}$
doublet. The underlying far-infrared (FIR) continuum samples the Rayleigh-Jeans
tail of the respective dust emission. By combining this information with our
earlier ALMA [CII]$_{\rm 158\mu m}$ observations, we explore the effects of
star formation and black hole feedback on the galaxies' ISM using the CLOUDY
radiative transfer models. We estimate dust masses, spectral indexes, IR
luminosities, and star-formation rates from the FIR continuum. The analysis of
the FSLs indicates that the [CII]$_{\rm 158\mu m}$ and [CI]$_{\rm 369\mu m}$
emission arises predominantly from the neutral medium in photodissociation
regions (PDRs). We find that line deficits are in agreement with those of local
luminous infrared galaxies. The CO spectral line energy distributions (SLEDs),
reveal significant high-$J$ CO excitation in both quasar hosts. Our CO SLED
modeling of the quasar PJ231-20 shows that PDRs dominate the molecular mass and
CO luminosities for $J_{\rm up}\le 7$, while the $J_{\rm up}\ge10$ CO emission
is likely driven by X-ray dissociation regions produced by the active galactic
nucleus (AGN) at the very center of the quasar host [abridged].
