Repeated games without a beginning and an end I'm interested to know results on repeated games indexing each stage game by $\mathbb Z$ which contrast with those indexed by $\mathbb Z_+$.

It seems to me this could be quite different from repeated games with a starting stage, because there's no need to concern ourselves about whether or not a node could be obtained. Is it the case that we only need to consider stationary equilibrium?
What you're asking seems to me just a matter of interpretation. Note that $\mathbb Z$ and $\mathbb Z_+$ have the same cardinality. So it makes no substantive difference which index set you use. 

Moreover, in a continuation game with finite history, any strategy can be interpreted as a (possibly non-stationary) Markov strategy in a continuation game with infinitely long history. 

If a history of infinite length is what you really care about, then may be looking into the literature of continuous-time repeated game will provide some insight.