We present a study of the connection among black hole accretion, star formation, and galaxy morphology at z ≤ 2.5. We focus on active galactic nuclei (AGNs) selected by their mid-IR power-law emission. By fitting optical to far-IR photometry with state-of-the-art spectral energy distribution (SED) techniques, we derive stellar masses, star formation rates, dust properties, and AGN contributions in galaxies over the whole COSMOS field. We find that obscured AGNs lie within or slightly above the star-forming sequence. We confirm our previous finding about compact host galaxies of obscured AGNs at z ∼ 1, and find that galaxies with 20%-50% AGN contributions tend to have smaller sizes, by ∼25%-50%, compared to galaxies without AGNs. Furthermore, we find that a high merger fraction of up to 0.5 is appropriate for the most luminous (log(LIR/L⊙) ∼ 12.5) AGN hosts and non-AGN galaxies, but not for the whole obscured AGN sample. Moreover, the merger fraction depends on the total and star-forming IR luminosity, rather than on the decomposed AGN infrared luminosity. Our results suggest that major mergers are not the main driver of AGN activity, and therefore obscured AGNs might be triggered by internal mechanisms, such as secular processes, disk instabilities, and compaction in a particular evolutionary stage. We make the SED modeling results publicly available.