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.
It helps if you consider the components of the acceleration of the smaller planets due to the gravitation force of each other planet. Here is a rough diagram showing the components of acceleration for each planet, assuming the largest does not accelerate due to its large mass:
The red arrow shows the component of acceleration of a planet due to the gravity of the largest planet. The green arrow is the component due to the other planet (not the largest).
Now, consider the bodies accelerate only by the red components of acceleration (i.e. ignore the gravitation effects between the two smaller planets). As the centres of gravity for the smaller planets are the same distance away, the motion of the two smaller planets will be perfectly symmetrical, and both planets hit the largest at the same time (assume smaller planets have similar radii/size).
Now, let us add the effects of the green arrows (i.e. the gravitation effect between the smaller planets). The angle between the red arrow and the green arrow is less than 90°, so this means that the green arrow will add to the acceleration of the planet towards to largest planet. Let us assume that the green arrows are sufficiently small so that the total component of acceleration perpendicular to the line connecting each small planet to the largest planet does not cause large rotational effects about the largest planet (i.e. effects such that the small planets do not travel in approx. straight lines towards the largest planet, complicating matters, and possibly throwing the smaller planets into orbit around the larger!).
The less massive planet will have a larger green arrow, and the larger of the two green arrows has a greater contribution to the component of acceleration in the radial direction (radial, as in the line connecting the smaller planet to the biggest planet). This means that, in this instance, the less massive planet is accelerating slightly faster than the slightly more massive planet towards the biggest planet. From this, it is sufficient to assume that... drumroll the smaller planet will hit the largest planet first!
This is true for all values of d and h, assuming the values don't cause the smaller planets to collide with each other first, and that the path between each smaller planet, and the largest planet is approximately straight. (i.e. d is above a critical value)
Best Answer
No. For example, the gravity of a cubical planet of uniform density, which can be computed analytically, is not directed towards its center (or any other single point).
You can also imagine a dumbbell-shaped mass distribution where the two heavy ends are very far apart. If you drop an apple near one end it is going to fall toward that end, not toward the middle of the “neck”.