Elliptic Curves, Lattices, Lie Algebras

1. No, for any cubic curve in the plane, there is a family of cubic plane curves (3 or 4 dimensional - I forget Edit: 8-dimensional, with a transitive action of PGL₃) that are isomorphic as curves. For any lattice in C, there is a 1-dimensional family of lattices that form isomorphic curves. Over C, the curves are classified by their j-invariant (a complex number).
2. You can fix the lattice and choose generators so that one of the generators is 1 and the other lies in the upper half plane. There is then an action of SL₂(Z) on the choices of generators, yielding an action on the upper half plane. If you choose a fundamental domain for this action, you get a "canonical" choice of lattice for each elliptic curve.
3. The Lie algebra is the unique 1-dimensional Lie algebra, whose bracket is zero. One can sometimes find more interesting information using the formal group law, but that mostly applies when you work in characteristic p.

What you want in terms of the relation between lattices and elliptic curves over C is proposition I.4.4 of Silverman's Advanced topics in the arithmetic of elliptic curves. Additionally, to go from a lattice to the equation of the elliptic curve (explicitly), you use Eisenstein series as in Corollary I.4.3 of that book. To go from the elliptic curve to the lattice is described in the proof of proposition I.4.4: you look at the homology of the curve and compute period integrals (incidentally, this is how you go from an abelian variety over C to a complex torus).

For a bit more info on question 3: if you are interested in the elliptic curve only as a complex Lie group, then when you identify it with C/L for C the complex plane and L a lattice, the Lie algebra is canonically C and the exponential map is the reduction mod L.