We present icarogw 2.0, a pure CPU/GPU python code developed to infer astrophysical and cosmological population properties of noisy, heterogeneous, and incomplete observations. icarogw 2.0 is mainly developed for compact binary coalescence (CBC) population inference with gravitational wave (GW) observations. The code contains several models for masses, spins, and redshift of CBC distributions, and is able to infer population distributions as well as the cosmological parameters and possible general relativity deviations at cosmological scales. We present the theoretical and computational foundations of icarogw, and we describe how the code can be employed for population and cosmological inference using (i) only GWs, (ii) GWs and galaxy surveys and (iii) GWs with electromagnetic counterparts. Although icarogw 2.0 has been developed for GW science, we also describe how the code can be used for any physical and astrophysical problem involving observations from noisy data in the presence of selection biases. With this paper, we also release tutorials on Zenodo.