Obviously, I haven't studied GR, I know no more than common knowledge. However, I'm wondering, is it impossible to develop a mathematical model based on flat space, in which the new equations of motion and gravity make the same predictions as GR? Or it's possible, but the equations will be very long and messy?
Newtonian physics is a special case of general relativity. It is described by the metric of flat space for special relativity where velocity is far less than the speed of light. In general, equations of motion in GR are described by the geodesic equation, which involves Christoffel symbols. These symbols describe the curvature of spacetime.