Elasticity of Space; How does the expansion of Space affect gravity? Does space have an elastic quality?
What I was thinking about was if space is expanding, is it being 'stretched', like a balloon being blown up, and if so, is this causing gravity to weaken? Imagine space as a 2 dimensional sheet (got this from one of Brian Greene's books) with planetary bodies resting on it and causing a depression in it, if you were pulling this sheet from all sides over a period of time, you would cause the depression of the planetary body to decrease and eventually become flat, which if we go back to reality, would mean that the gravitational 'constant' had changed to the point where the planetary body had no influence on those objects which were previously orbiting around it (or even residing on it's surface).
Is this the case in reality? Or does space not have an elastic quality? If not, can you explain to me what exactly it means for space to be expanding? 
In case you didn't notice, I'm a layman (hence the Brian Greene books :p), so try to keep your answers/explanations conceptual if possible.
 A: 
Does space have an elastic quality?

No it does not.1 You're taking the "rubber sheet" analogy too far. It's only meant to help you accept the fact that objects curve spacetime and that their motion is affected by this curvature. If you go any further into general relativity than that, the analogy breaks down. In particular, the rubber sheet analogy has nothing to say about the expansion of spacetime.
What it means for space to be expanding is simply that the distance between any two objects (which are not bound together by some force) grows with time. You can find further explanation of this point in another answer I've written and some of the other answers to that question.

1To be fair: perhaps somebody has proposed a theory about elasticity of space, but if so, the idea has not caught on.
A: I remember that Prof. Susskind said in the cosmology course of his "Stanford ongoing studies series" (it must have been somewhere in the first part of this 8 Lecture course)
http://www.newpackettech.com/Resources/Susskind/PHY28/Cosmology_Overview.htm
that space is continuously created in the course of the expansion such that the energy density keeps constant. This can be described by Hook`s law with a negative spring constant but it is not really a "rubber sheet". He mentioned to have derived a theoretical model to describe this process which is called "Newton Hooke cosmology".
A: But, how do gravitational waves propagate unless there were some form of fluidity to the space between bodies? There is a very fluidic like behavior of wht we refer to as the "fabric" of spacetime. Even with local large mass gravitational distortion, once the object moves from a given reference point the fabric returns to its previous contour like a boat through water. It may not be fluid by classical definitions, but...
A: It is my view point.....
Space is having a characteristic feature of elasticity which acts upon each fragment of universe both horizontally as well as vertically in a perfect balance or equilibrium. 
Space acts as something like elastic membrane that helps in keeping the planets in respective orbits.
Elastic fatigue of space- Universe started with a big bang that forced the detached fragments of a big mass to move outward randomly to a distance proportional to the mass of the fragment. The further movement of the fragments was restricted by the elastic force of the space. When the elastic fatigue will take place the fragments will start coming closer to a certain focal point of the elastic membrane of the space and it may again lead to collision of the planets and other bodies after losing elasticity.
There may be some other reasons for collision of celestial bodies. since, nothing lasts permanent, the Sun will also lose all its Hydrogen one day, what will happen then? It will swallow all other nearby bodies. 
