Complex fluids form a broad class of materials, examples of which include foodstuffs, personal care products, biological fluids, industrial compounds, and geological materials. Detailed knowledge of their flow behavior is therefore key to understanding and predicting natural phenomena or developing new materials. In the past century, major advances in both experimental and theoretical methods have enabled greater understanding of the connection between the microscopic physical structure of complex fluids and their material response. Traditional rheology involves imposing deformation on a macroscopic sample of fluid and measuring the stress response. However, an increasing need to interrogate microscopically-small fluid samples, and a heightened interest in the colloid-level physics of biology, have motivated the development of techniques to interrogate microliter-size samples and resolve micron-length heterogeneities. One of these experimental techniques is active microrheology, where a microscopically small probe particle is driven through the material of interest. Constitutive relations between the force required to drive the probe and the speed with which it moves allow one to infer the effective viscosity of the material, and more recent expansions of the theory of active microrheology allow the direct measurement of suspension stress by monitoring the mean and fluctuating motion of the probe. Theoretical models that connect probe motion to material and flow properties have enjoyed great expansion in the last decade for the study of complex fluids of particles interacting hydrodynamically, or with repulsive forces. However, many systems of interest, including biological cells, comprise colloidal suspensions that experience attractive forces not previously represented in theoretical models. This work presents a theoretical study of the impact of attractive forces on the microstructure, microviscosity, and nonequilibrium osmotic pressure of colloidal suspensions as measured by active microrheology. In active microrheology, the probe distorts the surrounding microstructure from its equilibrium configuration. The degree of this distortion is set by the strength of external forcing relative to entropic restoring forces; interparticle attractions and repulsions also influence this evolution. Although the effects of repulsions are well-studied in prior literature, a theoretical understanding of attraction-induced effects on nonequilibrium rheology is lacking. To examine how this interplay between different microscopic forces influences rheology, we formulate a Smoluchowski equation governing pair configuration as it evolves with flow strength, interparticle attractions, and hydrodynamic interactions. We determine its solution and compute microviscosity and nonequilibrium osmotic pressure from the structure via statistical mechanics. When the probe is subject to external forcing, attractions speed upstream probe-bath encounters and slow downstream detachment, transferring particle density downstream. This sets both the strength and direction of dipolar disturbance under weak forcing and the boundary-layer and wake structure under strong forcing. These attraction-induced structural changes affect rheology: both attraction-thinning and attraction-thickening are observed in the weak-forcing limit, while sufficiently strong forcing breaks attractive bonds and leads to flow-thinning or, in the presence of hydrodynamic interactions, flow-thickening. We find that the equilibrium osmotic pressure, described by the second virial coefficient B2, accurately predicts structural and rheological behavior in the weak-forcing limit regardless of specific attractive potential, but that the secondary length scale that arises under strong forcing precludes potential-agnostic observations far from equilibrium. The structural transitions and non-monotonic rheology in active microrheology show that tuning surface stickiness can either enhance or hinder probe motion and provide a means by which proteins or other macromolecules may change their surface chemistry to alter the viscosity of the surrounding medium, either speeding or slowing their own motion. Attractive forces are known to reduce the equilibrium osmotic pressure; as the second virial coefficient becomes more strongly negative, it is possible for phase separation to occur. This behavior changes away from equilibrium, where we find that the flow-induced nonequilibrium osmotic pressure reaches a minimum at a critical value of B2 before increasing with attraction strength. Hydrodynamic interactions suppress the nonequilibrium osmotic pressure and, at certain attraction strengths, can give rise to a flow-induced reduction in osmotic pressure below its equilibrium value. This denotes a flow-induced destabilization of attractive suspensions that may lead to phase separation in more concentrated systems, suggesting that self-assembly of active particles in biological suspensions may be driven by both attractive forces and hydrodynamic interactions.