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Just to give a few other examples of "non-obvious" equivalence between CS theories as MTCs, (G_2)_1 is the same as the even part of SU(2)_3 (maybe called SO(3)_3), and (E_8)_2 is the same as Spin(15)_1. There are probably many more.
How do you know the list of TQFTs in the table does not detect figure 8? The list is complete for Abelian gauge groups, so if it does not work then one has to go to non-Abelian gauge groups.
Kirby-Taylor ( www3.nd.edu/~taylor/old/papers/PSKT.pdf, see Lemma 3.6) describes how to compute an invariant for surfaces with Pin$^-$ structures, as the Gauss sum of a quadratic refinement on the first homology group. I think it is basically the $\eta$ invariant, in light of the relation to fermion SPT phases.
All the 16 modular tensor category with Ising fusion rules can be made unitary (so $d_\sigma=\sqrt{2}$) and the difference is only the twist. This MTC can be represented as conjugate of $(E_8)_2$, or the conjugate of $SO(15)_1$, so I doubt it has anything to do with the $E_8$ symmetry of Zamolodchikov.
There is a way to generalize Gauss-Milgram sum to Abelian fermionic topological order, which was discussed in arxiv.org/abs/1310.5708 (also arxiv.org/abs/hep-th/0505235). For the general case, although there is still a bulk-edge correspondence, the precise relation is not known.
Neveu-Schwarz and Ramond boundary conditions arise when you try to put the free fermion CFT on a torus, e.g. closed strings, and I think basically a choice of spin structures for Dirac spinors. For bosons there is no need to specify any spin structures, so just periodic boundary conditions. But a similar issue arises if you consider orbifolding of boson CFT.
