We present an extensive CO emission-line survey of the Plancks dusty GEMS, a small set of 11 strongly lensed dusty star-forming galaxies at z = 2-4 discovered with Planck and Herschel satellites, using EMIR on the IRAM 30-m telescope. We detected a total of 45 CO rotational lines from Jup=3 to Jup=11, and up to eight transitions per source, allowing a detailed analysis of the gas excitation and interstellar medium conditions within these extremely bright, vigorous starbursts. We applied radiative transfer models using the large velocity gradient approach to infer the spatially-averaged molecular gas densities, $n_{H_2}$~10$^{2.6-4.1}$ cm$^{-3}$, and kinetic temperatures, $T_k$~30-1000 K. In five sources, we find evidence of two distinct gas phases with different properties and model their CO ladder with two excitation components. The warm (70-320 K) and dense gas reservoirs in these galaxies are highly excited, while the cooler (15-60 K) and more extended low-excitation components cover a range of gas densities. In two sources, the latter is associated with diffuse Milky Way-like gas phases, which provides evidence that a significant fraction of the total gas masses of dusty starburst galaxies can be embedded in cool, low-density reservoirs. Finally, we show that the CO line luminosity ratios are consistent with those predicted by models of photon-dominated regions and disfavor scenarios of gas clouds irradiated by intense X-ray fields from active galactic nuclei. By combining CO, [CI] and [CII] line diagnostics, we obtain average PDR gas densities significantly higher than in normal star-forming galaxies at low-redshift, and far-ultraviolet radiation fields at least 100 times more intense than in the Milky Way. These spatially-averaged conditions are consistent with those in high-redshift SMGs and in a range of low-redshift environments.
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