Elasticity of Space; How does the expansion of Space affect gravity?
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.
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".