Substructures are ubiquitous in high resolution (sub-)millimeter continuum observations of circumstellar disks. They are possibly caused by forming planets embedded in the disk. To investigate the relation between observed substructures and young planets, we perform novel three-dimensional two-fluid (gas+1-mm-dust) hydrodynamic simulations of circumstellar disks with embedded planets (Neptune-, Saturn-, Jupiter-, 5 Jupiter-mass) at different orbital distances from the star (5.2AU, 30AU, 50AU). We turn these simulations into synthetic (sub-)millimeter ALMA images. We find that all but the Neptune-mass planet open annular gaps in both the gas and the dust component of the disk. We find that the temporal evolution of the dust density distribution is distinctly different of the gas'. For example, the planets cause significant vertical stirring of the dust in the circumstellar disk which opposes the vertical settling. This creates a thicker dust disk than disks without a planet. We find that this effect greatly influences the dust masses derived from the synthetic ALMA images. Comparing the dust disk masses in the 3D simulations and the ones derived from the 2D ALMA synthetic images, we find the former to be a factor of a few (up to 10) larger, pointing to that real disks might be significantly more massive than previously thought based on ALMA continuum images using the optically thin assumption and equation. Finally, we analyze the synthetic ALMA images and provide an empirical relationship between the planet mass and the width of the gap in the ALMA images including the effects of the beam size.