Please consider this before you move the motor mount. One of the big problems with outboards on the stern is that the motor is placed far back and as the boat pitches up and down in any significant chop, the engine will pop out of the water and you loose propulsion. In your situation, the motor is mounted relatively far forward and deep, so you should get the best propulsion forward in this position. Backing control, especially turning might improve a little by moving the engine further back so as to get a little more leverage, but if you are trying to back in a chop and the engine is popping out of the water more, you'll probably have a net loss.
When sailing, if the engine is in the water, and you leave the engine in neutral so the propeller can spin freely, and as long as the engine is placed on the boat's center line and not turned away from the centerline (acting like an supplemental rudder), I would not expect that you would get any difference in weather helm due to the engine. But if you were off center on the mount just a little, or the engine was turned away from center line, or the prop was locked in gear (the shape of the prop when not turning might also act as a supplemental rudder, requiring more tiller force to offset), then you could get weather helm. Some might say letting the prop spin freely is hard on the gearing....perhaps so, but Yanmar has sent out a bulletin on my engine saying that it's better to let the prop free wheel, otherwise there might be some transmission damage resulting when in gear while sailing with engine not running.