We present imaging and spectropolarimetric observations of the ultraluminous infrared galaxy IRAS P09104+4109 using the Keck 10 m Telescope. We detect the clear presence of broad H beta, H gamma, and Mg II lambda 2800 emission lines in the polarized flux spectra of the nucleus and of an extranuclear emission region similar to 4 " away, confirming the presence of a hidden central quasar. The polarization of the broad Mg II emission line is high (similar to 29%), consistent with the remarkably high polarization (similar to 30%-40%) observed in the extended continuum emission. This indicates that the off-nuclear continuum is dominated by light scattered from the hidden quasar, most probably by dust mixed with the line-emitting gas. The high polarizations, combined with the "foreshortened" morphology of the polarized brightness distribution, allow us to constrain the scattering biconical structure to be at inclination i approximate to 50 degrees with a half-opening cone angle theta c approximate to 40 degrees. The narrow emission lines are polarized in a stratified fashion, with the high-ionization lines ([O III], [Ne V], [Fe VII]) being polarized 0.7%-1.7% and [O II] essentially unpolarized. The line polarizations are positively correlated with critical density, ionization potential, and velocity width of the emission lines. This indicates that, as is the case with the narrow-line radio galaxies, which also often contain powerful quasars, the narrow emission line region may be partially shadowed by the putative torus, with the higher ionization lines originating closer to the nucleus. One notable characteristic of the extranuclear knot is that all species of Fe are markedly absent in its spectrum, while they appear prominently in the nucleus. In addition, narrow Mg II is observed to be much weaker than predicted by ionization models. Our favored interpretation is that there is a large amount of dust in the extranuclear regions, allowing gaseous refractory metals to deposit. Near the nucleus, dust is destroyed in the strong radiation field of the quasar, inhibiting metal depletion onto grains. The extended emission regions are most likely material shredded from nearby cluster members and not gas condensed from the cooling flow or expelled from the obscured quasar. The higher temperature inferred from CO mi lines compared to that from [O III] and the general better agreement with models of line ratios, especially CO mi lambda 5007/lambda 4363 and He II/H beta, provide strong evidence for matter-bounded clouds in addition to ionization-bounded clouds in the narrow-line region. Ionization by pure velocity shocks can be ruled out. Shocks with photoionizing precursors may be present but are probably not a dominant contributor to the energy input.