We present a study of the structure of Co/W(001) determined by low-energy electron diffraction, the film stress measured by a bending thin substrate, and the film morphology measured by scanning tunneling microscopy in combination with in situ magnetic characterization by the magneto-optic Kerr effect. Different regimes in the stress behavior during growth are correlated to surface effects and structural and morphological changes in the Co film. Initially a compressive stress is observed as the result of the lower surface free energy of Co compared to W, followed by a tensile stress due to the large misfit between film and substrate. The structural transition from bcc to hcp Co as well as dislocation formation is clearly observed in the stress measurements and by low-energy electron diffraction and scanning tunneling microscopy. Additionally, the morphology of the films is correlated with the magnetic properties and a new surface alloy at submonolayer coverages is reported.