The topological properties of many materials are central to their behavior, with the dynamics of topological defects being particularly important to intrinsically out-of-equilibrium, active materials. In this paper, local manipulation of the ordering, dynamics, and topological properties of microtubule-based extensile active nematic films is demonstrated in a joint experimental and simulation study. Hydrodynamic stresses created by magnetically actuated rotation of disk-shaped colloids in proximity to the films compete with internal stresses in the active nematic, enabling local control of the motion of the +1/2 charge topological defects that are intrinsic to spontaneously turbulent active films. Sufficiently large applied stresses drive the formation of +1 charge topological vortices in the director field through the merger of two +1/2 defects. The directed motion of the defects is accompanied by ordering of the vorticity and velocity of the active flows within the film that is qualitatively unlike the response of passive viscous films. Many features of the film's response to the disk are captured by Lattice Boltzmann simulations, leading to insight into the anomalous viscoelastic nature of the active nematic. The topological vortex formation is accompanied by a rheological instability in the film that leads to significant increase in the flow velocities. Comparison of the velocity profile in vicinity of the vortex with fluid-dynamics calculations provides an estimate of film viscosity.