Molecular gas is the key ingredient of the star formation cycle, and tracingits dependencies on other galaxy properties is essential for understandinggalaxy evolution. In this work, we explore the relation between the differentphases of the interstellar medium (ISM), namely molecular gas, atomic gas, anddust, and galaxy properties using a sample of nearby late-type galaxies. Tothis goal, we collect CO maps for 121 galaxies from the DustPedia project,ensuring an accurate determination of $M_{H2}$, the global molecular gas mass.We investigate which scaling relations provide the best description of$M_{H2}$, based on the strength of the correlation and its intrinsicdispersion. Commonly used correlations between $M_{H2}$ and star formation rate(SFR) and stellar mass ($M_{\star}$) are affected by large scatter, whichaccounts for galaxies that are experiencing quenching of their star formationactivity. This issue can be partially mitigated by considering a "fundamentalplane" of star formation, fitting together $M_{H2}$, $M_{\star}$, and SFR. Weconfirm previous results from the DustPedia collaboration that the total gasmass has the tightest connection with the dust mass and that the molecularcomponent also establishes a good correlation with dust. Although dust grainsare necessary for the formation of hydrogen molecules, the strength ofgravitational potential driven by the stellar component plays a key role indriving density enhancements and the atomic-to-molecular phase transition.Eventually, we investigated the correlations between ISM components andmonochromatic luminosities at different wavelengths: we proposed mid and far-IRluminosities as reliable proxies of $L^{\prime}_{CO}$ for sources lackingdedicated millimeter observations. Luminosities in mid-IR photometric bandscollecting PAH emission can be used to trace molecular gas and dust masses.

(Abridged) The relations between stellar ($M_\ast$), gas ($M_{\rm gas}$),baryonic ($M_{\rm bar} = M_\ast + M_{\rm gas}$), and dark matter halo mass($M_{200}$) provide unique constraints on galaxy formation and cosmology. Theshape of the relations constrains how galaxies regulate their growth throughgas accretion, star formation, and feedback; their scatter probes thestochasticity of galaxy assembly. Here, we assemble a sample of 49 nearby gas-rich dwarf and massive discgalaxies with unmatched ancillary data. We obtain their gas kinematics andderive their dark matter properties through rotation curve decomposition. Oursample allows us to study the galaxy-halo connection across nearly six ordersof magnitude in $M_\ast$. We find that the $M_{\rm gas}-M_{200}$ relation risesmonotonically, with galaxies having around 4 per cent of the averagecosmological baryon fraction in cold gas. Contrastingly, the $M_\ast-M_{200}$relation shows a more complex behaviour. A particularly interesting finding isthat of a population of baryon-deficient' dwarfs (BDDs) with stellar masses$\sim 1-1.5$ orders of magnitude lower than expected from current models. Yet,baryon-rich galaxies also exist, and we find a large spread in the baryonretention fraction across our galaxies. We compare our findings withsemi-analytic and hydrodynamical galaxy formation simulations. While thesimulations broadly reproduce most observed features, they struggle to matchthe BDDs and do not capture the diversity in baryon fractions. Understandingthese differences will shed new light on how feedback regulates galaxyformation. Finally, we study the dark matter halo concentration-mass relation.We find that below $M_{200} \sim 10^{11}\,M_\odot$, the concentrations aresystematically lower than expected. We discuss whether these results stem fromthe influence of baryonic physics or the environment.