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]$_{rm 205mu m}$, [CI]$_{rm 369mu m}$, 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$_{rm 163mu m}$ 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]$_{rm 158mu m}$ 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]$_{rm 158mu m}$ and [CI]$_{rm 369mu m}$ 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].
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