Does solar frame dragging induce a higher centrifugal force to Mercury so its orbit is precessing as well as being more outwards than it should be if there wasn't solar frame dragging?
Frame dragging is not responsible for Mercury's precession, that can be explained with a combination of Newton's planetary tug and the Schwarzschild metric. Frame dragging has the effect that the a prograde circular orbit at a given radius need smaller velocity than without frame dragging, and a larger velocity for a retrograde orbit. The perihelion is also shifted a bit, but that is not significant for Mercury. In the strong field around a black hole the effect is more visible: on the left the orbit is around a Schwarzschild black hole and therefore without frame dragging, and on the right around a Kerr black hole. Both start with the same initial conditions at x=20 with v=5/12/√2:
Since they both start with the same initial velocity and at the same distance, but the one on the right has frame dragging for support, its farthest radius is farther than the one without frame dragging.