We investigate the structure and theological properties of dilute colloid-polymer mixtures at rest and under shear via molecular simulations that take into account hydrodynamic interactions. Mixtures of amphiphilic Janus colloids (JCs) and hydrophobic/amphiphilic polymers are considered for various solvent qualities and polymer concentrations. Free polymers, small polymer droplets, and hybrid aggregates coexist in mixtures with slightly hydrophobic homopolymers. As the solvent quality worsens, all polymers aggregate into small droplets, covered and stabilized by the JCs. In mixtures with amphiphilic polymers, we observe the coexistence of free polymers, purely polymeric micelles, and hybrid aggregates. At low shear rates, all mixtures exhibit a Newtonian-like response with intrinsic shear viscosities that are up to 2 times as large as of pure suspensions of nonadsorbing colloids at the same concentration. Furthermore, the mean aggregation number increases slightly due to the flow-enhanced collision of aggregates. At larger shear rates, however, the aggregates break up, the polymers align in the flow direction, and the mixtures exhibit shear-thinning. This shear-induced breakup occurs at stronger shear compared to pure JC suspensions, indicating that the adsorbed polymers reinforce the hybrid aggregates.