Aims. We present and study spatially resolved imaging obtained with the Atacama Large Millimeter/submillimeter Array (ALMA) of multiple ¹²CO(J =6-5, 8-7, and 9-8) and two H₂O(2₀₂-1₁₁ and 2₁₁-2₀₂) emission lines and cold dust continuum toward the gravitationally lensed dusty star-forming galaxy SPT 0346-52 at z = 5.656. Methods. Using a visibility-domain source-plane reconstruction we probe the structure and dynamics of the different components of the interstellar medium (ISM) in this galaxy down to scales of 1 kpc in the source plane. Results. Measurements of the intrinsic sizes of the different CO emission lines indicate that the higher J transitions trace more compact regions in the galaxy. Similarly, we find smaller dust continuum intrinsic sizes with decreasing wavelength, based on observations at rest frame 130, 300, and 450 μm. The source shows significant velocity structure, and clear asymmetry where an elongated structure is observed in the source plane with significant variations in their reconstructed sizes. This could be attributed to a compact merger or turbulent disk rotation. The differences in velocity structure through the different line tracers, however, hint at the former scenario in agreement with previous [CII] line imaging results. Measurements of the CO line ratios and magnifications yield significant variations as a function of velocity, suggesting that modeling of the ISM using integrated values could be misinterpreted. Modeling of the ISM in SPT 0346-52 based on delensed fluxes indicates a highly dense and warm medium, qualitatively similar to that observed in high-redshift quasar hosts.