Is a 2-D periodic structure isomorphic with the surface of a torus, a sphere, neither or both?
Yes, a 2D-periodic space can be mapped to a torus, but that's more a question for the math.SE…

Regarding your bonus question, why would there be? What would you do with it? Molecular structures are intrinsically 3D, so I don't see what you would do in a 2D (periodic or not) space? Even when we talk about planar or pseudo-2D structures (buckyball, nanotube, etc.) they are 3D objects with 3D electronic densities and wavefunctions.



Edit: 3D structures that are periodic in two dimensions and finite in the other one can be studied by many computational chemistry codes. They are often referred to as slab calculations or surface calculations. The most common issue is that of Coulombic interaction (or Poisson equation solver), which typically requires special treatment in the 2D case.