I will keep things simple. Keep in mind gravity tends to pull everything together and never repels objects. Gravitational field of the object depends on the shape and the matter distribution. A deformed planet with the same mass as that of the Earth will behave just like the earth from a distance. The non-spherical effects will become more obvious on approaching the object.
Due to lack of spherical symmetry, objects will definitely accelerate with different values and directions around the surface object.
Things would definitely have different weights in different regions on the surface.
Gravity wants to smash everything together into smallest size possible, only non-gravitational forces can cause repulsion.
For the slingshot question, it depends on many factors that are both unique to the object and that are same for all planets.
If you are too far from the planet, you will not be able to slingshot. If you are moving too fast, you will not be able to slingshot. But if you get your initial approach distance and velocity just right, you will be able to slingshot around the non-spherical planet. There are many combinations of the two that could make you slingshot.
The translational motion of any free rigid object can be analyzed by looking only at the motion of its center of mass. We can thus concentrate the whole mass of the object at a point. Basically, the orbit is the path traced by this point (the center of mass).
So if the non-spherical planet is rigid enough, it will trace a uniform path around the Sun (an ellipse). The shape of the ellipse will in general depend on the mass of the object and its energy. So the orbit will in general be a squashed or stretched variation of Earth's orbit.
Remember the Earth was not always shaped like a ball (well almost). It became a big ball over billions of years. However, if the object is rigid enough it will not turn into a ball.
This is off topic, but having read the books and seen the movie, I can tell you that the former are much much better than the latter.