I will attempt to answer with very little string theory background - because your questions seem oriented towards this basic case rather then the theory in general.
1. First, a correction. On page 7 of that article, it defines $A=iR_1R_2$, not $R_1/R_2$. So, since the torus is flat, $A$ is $i$ times the usual area $R_1R_2$.
2. As you say, a complex structure is a map $J$ such that $J^2=-1$. It comes from thinking of the complex structure on $\mathbb{C}$, where $iz=i(x+iy)=-y+ix$, so it exchanges the roles of the two coordinates. If the torus is a rectangular region of $\mathbb{C}$ with the opposite sides identified, the complex structure is a "rotation+flip", and changes the appearance of the rectangle. Since $\tau$ is the ratio of the two sides of the rectangle, it tells us something about the shape of the torus [some clarity below].
3. The torus is a CY manifold in 1 dimension, so the symmetry $A\leftrightarrow\tau$ is a map between two CY manifolds. 586
He equates this with T-duality $A\leftrightarrow 1/A$, which is closely related to mirror symmetry.
4. Well, to be clear we are talking about the torus metrics, which are completely specified by $R_1$ and $R_2$. (This is not the same as "the moduli space of $\mathbb{T}^2$" because that would mean more or less structure, depending on the context. To a topologist, the moduli space of tori is 0-dimensional, since there is only one 2d topological surface with genus 1). That means just $A$ wouldn't cut it - there would be pairs $(R_1,R_2)$ with the same $A$ but different sizes. If you include $\tau$ (linearly independent from $A$), then you can break that degeneracy. So the moduli space is parametrized by either the pair $(R_1,R_2)$ or $(A,\tau)$. (He does say that for more general tori you need to consider real parts for $A$ and $\tau$, so the moduli space would be bigger).
[Some Clarity] In case that wasn't clear - consider the complex structure of $\mathbb{C}$, the imaginary unit $i$. It's action on the edges is
$(R_1,R_2)\rightarrow (-R_2,R_1)$
So what happens to $A$ and $\tau$ under this map?
$$A=iR_1R_2\rightarrow A'=-iR_2R_1$$ $$\tau=iR_2/R_1\rightarrow \tau'=i(R_1)/(-R_2)$$
So $A$ doesn't tell us anything about the complex structure, because under that map we just get $A\rightarrow -A$. However, $\tau\rightarrow -1/\tau$, so $\tau$ tells "how wide" and "how long" the torus is (at least, the ratio of these), which is the complex structure. 364