We present a multiline survey of the interstellar medium (ISM) in two z >6 quasar 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 11 transitions including atomic fine-structure lines (FSLs) and molecular lines: [NII](205 mu m), [CI](369 mu m), CO (J(up) = 7, 10, 15, 16), H2O 3(12) - 2(21), 3(21) - 3(12), 3(03) - 2(12), and the OH163;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](158 mu m) observations, we explored the effects of star formation and black hole feedback on the ISM of the galaxies using the CLOUDY radiative transfer models. We estimated dust masses, spectral indexes, IR luminosities, and star-formation rates from the FIR continuum. The analysis of the FSLs indicates that the [CII](158 mu m) and [CI](36 mu m) emission arises predominantly from the neutral medium in photodissociation regions (PDRs). We find that line deficits agree with those of local luminous IR 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(up) <= 7, while the J(up) >= 10 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. The J(up) >10 lines are undetected in the other galaxies in our study. The H2O 3(21) - 3(12) line detection in the same quasar places this object on the LH2O - L-TIR relation found for low-z sources, thus suggesting that this water vapor transition is predominantly excited by IR pumping. Models of the H2O SLED and of the H2O-to-OH163 mu m ratio point to PDR contributions with high volume and column density (n(H) similar to 0.8 x 10(5) cm(-3), N-H = 10(24) cm(-2)) in an intense radiation field. Our analysis suggests a less highly excited medium in the companion galaxies. However, the current data do not allow us to definitively rule out an AGN in these sources, as suggested by previous studies of the same objects. This work demonstrates the power of multiline studies of FIR diagnostics in order to dissect the physical conditions in the first massive galaxies emerging from cosmic dawn.