Diblock polymer micelles dispersed in an aqueous environment are being actively investigated for various applications, but there is only a qualitative understanding of the effect of the chemical structure on the micelle hydration and water dynamics as these properties are difficult to assess experimentally. Using all-atom molecular dynamics simulations, we investigate aqueous solutions of three comparable in size diblock copolymer micelles with core-forming blocks of different hydrophobicity: polybutadiene (PB), polycaprolactone (PCL), and polytetrahydrofuran (pTHF) with the same hydrophilic block, polyethylene oxide (PEO). We found that core-block hydrophobicity and ability to form hydrogen bonds with water strongly affect the water dynamics near the core: water molecules spend considerably less time in contact with the PB block than with PCL and pTHF blocks. We obtained polymer and solvent volume fraction profiles and determined that the interfacial width systematically increases with a decrease of core block hydrophobicity with water penetration into the core being negligible for PB-PEO and PCL-PEO micelles, while for pTHF-PEO micelles the interface is more diffuse and there is a noticeable penetration of water (17% by volume). For PCL-PEO micelles, which are commonly used in biomedical applications, we also investigated tetrahydrofuran (THF) penetration into the micelles from mixed THF:water solution at early stages of micelle dissolution. We found an inhomogeneous solvent distribution with a maximum of THF volume fraction in the interfacial core-corona region and partial exclusion from the PEO corona, which slows down micelle dissolution. These results can have important implications for micelle stability and use in biomedical applications.