Angelo,
Does it all reach us and "intermingle" or does the strongest simply "cancel" out the other...
(If it did, would our gravitional pull on the moon, not cancel out the effect of it's on ours?)
The upshot of what you are describing is called "the generalized n-body problem" and it has never been solved. ("n" refers to the number of bodies that you are trying to resolve gravitational forces for) But in essence, yes, you are correct that all bodys exert gravitaional force on others with respect to the "universal" law of gravitation:
http://en.wikipedia.org/wiki/Newton's_law_of_universal_gravitation
This article describes the formulation of this law for only two bodies. It gets much more complex when you add a 3rd body (but not unsolvable). When you add 4 or more bodies, well, as far as I know no one has come up with an exact solution for n>=4.
Is there any way to identify individual gravitional effect once it reaches here, and if so, control each individually through any means? (artificially cancel one or more out?).
I think you will probably see from the answer above (n-body problem with n>=4) that we currently have no means to individually isolate each body's effect while computing the effects of all other bodies. In general, we tend to solve such equations by only considering the closest and most massive bodies in a given region for a given astrodynamics problem (such as navigating a probe from earth to Mars such that it intercepts the red planet).
However, one of the things that you may be referring to when you talk about "artificially cancel one or more out" would be the Lagrange points that exist between any two bodies:
http://en.wikipedia.org/wiki/Lagrange_Points
As shown there are five such points for any 2-body situation where, essentially, the gravity of the two bodies is in-balance. Theoretically, an object that exists at any of these points would remain there forever without becoming disturbed and gravitating to either of the two bodies. But in reality, since gravity actually IS an "n-body problem" an object does not forever stay steady at a Lagrange point. Instead, we realize that if I place an object at a Lagrange point, this is the minimum energy position where I would have to expend the smallest amount of propellant to keep the object stable at that point. Essentially, to keep the object there, every so often I would have to fire thrusters to get it back on point. These thruster firings are compensating for the gravity effects of other bodies in the universe inducing error in the 2-body problem. A quote from the above article relates Lagrange points to the n-body problem:
<font color="red"> "A more precise but technical definition is that the Lagrangian points are the stationary solutions of the circular restricted three-body problem." [/COLOR]
RMT
