Gravity is at least primarily based on mass, more mass = more gravity. Planets and stars are spherical because gravity is great enough to compress them into spherical shapes, but (at least as I understand it) even stars are not perfect spheres. As well, most planets are not made of just one element or molecule, so there is also different amounts of matter in different areas of the planet. Smaller objects might be able to be purely one element or molecule, but are usually too small to become spherical due to gravity. Because of this, mass is not perfectly distributed evenly, and so neither is gravity. This can and has been calculated on earth. 9.8 M/s is just the rounded average. Wikipedia's article notes the gravity varies from 9.76 to 9.83 M/s on the surface depending on location.
Since Gravity is not exactly the same all around, then calculating the net force half the planet exerts, and then comparing it to the other half, the forces should not be exact opposites of each other for almost every single plane you choose (With very few exceptions, though probably at least one thanks to the Intermediate Value Theorem). While on earth the mass distribution can change slightly, its so big its probably going to be negligible, while on asteroids, they mostly will not change ever without external forces. Overall, once a net force is found, it would in most instances exist for at least a very long time.
1) Can this net force actually exist due to the planets gravity?
2) If it does exist, Could this force cause the object to move? If not, Why does it not exist, or what cancels it out?
If necessary, Assume there would be some way to tell that it is moving and/or accelerating.