I believe that on the first two questions, there is no answer (so they are not well-posed:-)
The reason is that the flat structure on a torus and the moduli spaces of tori were very
well understood long before the notions of "flat", "structure", "moduli space" and even "tori"
were introduced. Mathematicians just did not think in these terms.
Once these notions were introduced, the tori became a simple example.
Teichmuller was probably a person who made the notion of
"moduli space" completely rigorous, though it was considered for the first time by Riemann.
And Teichmuller considers
tori at a great length to explain what he means. But certainly he did not find anything
new about tori. This was just the simplest example.
(Many people think, perhaps correctly, that Abel and Jacobi already knew
everyhting about tori, though they did not use this word). Modular group was known to Gauss
at about the same time. It is true, these things were developed for another 70 years or so,
culminating in the Fricke-Klein and enormous French "treateases".
But I know people who still prefer Jacobi's exposition, and even study Latin to read it:-)
The metric of constant negative curvature on surfaces of genus >1 is another story.
This was introduced by Poincare. Even the Poincare metric on the unit disc was apparently
(Schwarz lemma was stated and proved much later, and not by Schwarz:-)
About the last question, I suppose Klein has to be credited. It was Klein who in
his "Erlangen program" explained explicitly "what geometry is", and how it is connected with a group
action of certain kind.
Klein was in close collaboration with Lie at that time.
There is an excellent "primary source" on how all these things really developed:
Klein's Lectures on the history of Math in XIX century.