Simple current extensions in VOA theory and CFTs

I'm not a physicist, so I can't say anything authoritative on your first question, but I can say something about questions 2. and 3.

You are correct that in VOA theory, simple currents are invertible irreducible objects in a vertex tensor category of representations of a VOA (although if one does not know that the vertex tensor category structure exists, one can still define simple currents in terms of intertwining operators among modules in the representation category under consideration). A simple current extension essentially amounts to the action of a compact abelian group on a VOA. More precisely, given a compact abelian automorphism group $G$ of a simple VOA $V$, $V$ is a simple current extension of the subalgebra $V^G$ of $G$-fixed points, at least as long as $V^G$ admits a suitable representation category with vertex tensor category structure (this is proved in https://arxiv.org/abs/1603.05645 and https://arxiv.org/abs/1511.08754). Conversely, if $V$ is a simple current extension of some subalgebra, then $V$ is naturally graded by the (discrete) abelian group $G$ of simple currents and hence admits an action of the compact dual group of characters $\widehat{G}$.

With this perspective, one can see that simple current extensions are a quite restricted class of VOA extensions. For example, non-abelian automorphism groups lead to extensions $V^G\subset V$ which are not of simple current type: the tensor category structure on the irreducible $V^G$-modules occurring in $V$ will reflect the tensor structure of the category of finite-dimensional $G$-modules (this is shown in my recent paper https://arxiv.org/abs/1810.00747). And most VOA extensions do not arise from an automorphism group at all.

However, simple current extensions have received a lot of attention in the VOA literature, probably because they are easier to study than more general extensions, but also because a lot of interesting VOAs can be constructed as simple current extensions of some subalgebra. Perhaps the first such example is the celebrated moonshine module $V^\natural$ of Frenkel, Lepowsky, and Meurman, which is an order-$2$ simple current extension of $V_{\Lambda}^+$, the fixed-point subalgebra of the Leech lattice vertex operator algebra $V_{\Lambda}$ under an automorphism induced from the $x\mapsto -x$ isometry of the Leech lattice $\Lambda$ (although the original construction did not use simple current extension techniques to prove that $V^\natural$ is a VOA). More recently, the generalization in https://arxiv.org/abs/1507.08142 of the Frenkel-Lepowsky-Meurman construction of $V^\natural$ has contributed to the program of constructing and classifying all holomorphic VOAs of central charge $24$.

For additional references on simple current extensions, you can consult the reference lists of some of the above papers. Another nice recent paper is https://arxiv.org/abs/1711.05343 which deals with infinite-order simple current extensions from a tensor-categorical point of view. This paper also references a number of earlier papers on simple current extensions.

Regarding your first question, physicists are interested in classifying modular invariant partition functions for two-dimensional rational conformal field theories. Simple currents are a useful tool for constructing such partition functions. An early physics paper on the subject that develops this point of view is https://www.sciencedirect.com/science/article/abs/pii/0370269389909489?via%3Dihub

Furthermore, this paper also introduced the name "simple current".