I don't have an answer to your other questions, but I can address the first question. The vertex algebra for the rank $d$ free boson is (after forgetting some structure) a graded vector space whose weight 1 subspace is naturally a $d$-dimensional abelian Lie algebra. In this setting (in particular, because of the "freeness"), modules for the vertex algebra are in natural bijection with representations of this Lie algebra. Irreducible representations are nicely parametrized by points in $d$-dimensional space, but the extensions can be a bit of a nightmare. There is a monoidal structure on modules, which amounts to addition for the parameters attached to irreducible modules.

In conclusion, this particular category is more tractably written as "representations of an abelian Lie algebra", or equivalently, "representations of a polynomial ring".

I don't have an answer to your other questions, but I can address the first question. The vertex algebra for the rank $d$ free boson is (after forgetting some structure) a graded vector space whose weight 1 subspace is naturally a $d$-dimensional abelian Lie algebra. In this setting (in particular, because of the "freeness"), modules for the vertex algebra are in natural bijection with representations of this Lie algebra. Irreducible representations are nicely parametrized by points in $d$-dimensional space, but the extensions can be a bit of a nightmare. There is a monoidal structure on modules, which amounts to addition for the parameters attached to irreducible modules. There is also a braiding coming from a quadratic form that we fix in the beginning, but let us ignore that for now.

In conclusion, this particular category of modules is more tractably written as one of the following:

1. finite dimensional representations of an abelian Lie algebra
2. finite dimensional representations of a polynomial ring
3. coherent sheaves of finite length on affine space