First part: I want to understand intuitively when two complex structures on a torus agree. Is it true that it all just comes down to the fundamental lattices being similar, i.e., having the same angles and proportional side lengths? This seems reasonable (since angles must be preserved in holomorphic transformations), but I think that I have never heard it stated like this. If it is true, then I think that we could wlog scale and rotate the lattice generators so as to have $(1,\tau)$ with $\tau$ in the open upper half-plane. Then we take $0<\arg(\tau)\le \pi / 2$ and $0 < \vert \tau \vert \le 1$ and this should classify all complex structures by choosing the smallest angle and the side length ratio for the fundamental domain. But I have a feeling that this is not correct...
Second part: is there a way to concretely construct the torus inside $\mathbb{R}^3$ such that the angles measured there with the usual euclidean metric correspond to the ones measured in the fundamental parallelogram? This seems unlikely, because the usual gluing process to construct the torus from the parallelogram distorts angles and lengths...
COMMENT.- Anyway you do have that $\mathbb C/L$ is isomorphic to the product $\Gamma\text{ x }\Gamma$(bidimensional torus) where $\Gamma=\{z\in\mathbb C: |z|=1\}$ (which is a multiplicative group) and $L$ is your lattice. This is because $L$ is the kernel of the homomorphism onto from $\mathbb C$ to $\Gamma\text{ x }\Gamma$ defined by $$\alpha_1w_1+\alpha_2w_2\to(e^{2\alpha_1\pi},e^{2\alpha_2\pi})$$ which is easily verified.