How does the gravity of a massive non-spherical object act on things around it? Firstly, not sure if this question ought to be in the space SE site. Please let me know if it should. (Posted in both for now)
Secondly, I don't know a whole lot about physics (I'm just inquisitive). So please try to keep answers simple (or at least dummed down to laymans terms).
Inspired by "A Hitchhikers Guide to the Galaxy" movie (haven't read the book yet)
My question is, for a non-spherical object (lets say a cube for now) of a size and mass like that of Earth, how would objects act in it's gravity? 
Aspects of this question lead to the following questions
•Would something weigh more/less in certain areas over other areas?
•I'm aware that it's possible to 'slingshot' around massive spherical objects to change their directory. Would it be possible to slingshot objects around something non-spherical to change their velocity?
•If this massive object was orbiting our sun, would it's orbit be as uniform as ours? if not what would that look like and why?
 A: 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.
A: This is a 1 million dollar question! When I studied Celestial Mechanics, I learnt that celestial bodies taken together mainly by gravitational forces must be spheroidal or, at most, ellipsoidal, if the body is spinning. This comes from the inverse-square law for the gravitational force and from the fact that gravity is always attractive. With this 'simplified' massive bodies (with respect, say, to a giant cube!), the problem of the orbit of a satellite can be analytically solved, often with further simplifications. Anyway, the net effect, with respect to a Keplerian elliptic orbit, is the presence of an apsidal precession that makes the elliptic orbit 'rotate'.
Any other complication to the motion, such as other exotic shapes, can only be solved by approximations in the mass multipole moments. Then, I really can't figure it out what could it be the motion of a satellite near a giant cubic planet!
A: It turns out that the mass of bodies such as the Earth or the moon is not perfectly evenly distributed; anomalous mass concentrations or mascons actually do have to be taken into account when planning e.g. moon landings. At a great distance from an irregular massive object, you can treat it as a point mass, but closer in, the irregularities matter. 
