It is my understanding that the original motivation did come from GW theory: Chen and Ruan came across the orbifold cohomology ring, and its product, as a byproduct and first step of defining the orbifold quantum cohomology.  The strongest motivation I have for the grading also comes from this perspective, and maybe we have to start with the question of why Chen and Ruan were trying to define orbifold GW theory to begin with.

The fairy tale story I have in my head is that string theorists found that they could do string theory on mildly singular spaces.  Often, rather than dealing with singularities, we instead get rid if it, by either a) resolving the singularity, or b) deforming the equations to something smooth. If our singular space is actually a smooth as an orbifold, orbifold cohomology gives us a third option, c).

Now, the physicists also had intuition that, in nice cases and if we look at it the right way, all three approaches should essentially give us the same thing.  So, for instance, passing from a) to b) directly can go by the name "conifold transition".

To get back to the age grading: a) and c) are related by what has come to be known as the Crepant Resolution conjecture: if $f:Y\to X$ is a crepant resolution of an orbifold $X$ (i.e. $f$ is a resolution and $f^*(K_X)=K_Y$), then $X$ and $Y$ should have "the same" quantum cohomology.  In particular, the base graded vector spaces of $X$ and $Y$ should be the same.  

A strong motivation for the age grading of quantumn cohomology is that it makes this true: if $X$ is an orbifold that has a crepant resolution, then the graded vectorspace $H^{orb}(X)$ is isomorphic to the graded vector space $H^*(Y)$, for any crepant resolution $Y$.  This statement about the graded vector space was proved using motivic integration by [Yasuda][1], and independently by Lupercio and Poddar, and provides a lot of evidence that the age shifting is the "right" shifting.  I should note that the stronger statement about the orbifold quantumn cohomlogy rings being "the same" is still open, and in the general case requires some work to even state correctly, as in this [answer][2] by Jim Bryan.

I feel like a really good answer to your very last question should be possible and not that hard, but I don't have it at my fingertips now.  So I'll just give a short "yes": the obstruction bundles used to define the orbifold cup product are exactly cohomology spaces of bundles on orbifolds, and so the RR formula is going to pop up there.  If you're okay with the age grading showing up in the RR formula, then from the definition of the quantum product it follows that the age grading of $H^{orb}$ has to be what it is.  I feel like I'm glib and just sort of restating your question -- I'm trying to say I think you're right.



 


  [1]: http://arxiv.org/abs/math/0110228
  [2]: https://mathoverflow.net/questions/47322/quantum-cohomology-rings-as-invariants/47352#47352