It's certainly used in the paper by Bao. I didn't see how the other papers use it to prove their results about band sums, although Eudave-Muñoz does invoke it afterward to apply his result to the cabling conjecture.

Done, unless you had any other developments in mind.

@nikita: The adjunction inequality says that $tb(L)\le 2g_*(L)-1$, so equality is still possible, but either way if you have a symplectic surface then it actually satisfies an adjunction formula $\langle c_1(\omega), \Sigma\rangle + \Sigma\cdot\Sigma = 2g(\Sigma)-2$. In response to your second comment, if the surface is symplectic then its branched double cover is naturally a symplectic manifold and so the contact structure which comes from taking the branched double cover of the transverse knot in S^3 is in fact symplectically fillable.

The 3-torus can't be a branched double cover because the triple cup product on $H^1$ would then be zero.

See "Approximating C^0-foliations" by Kazez and Roberts (arxiv.org/abs/1404.5919), which addresses exactly this issue.

There are actually lots of non-QA knots whose branched double covers are L-spaces. See section 6.1 of arxiv.org/pdf/1205.5261.pdf for some discussion and explicit examples, including the $P(p_1,\dots,p_n,-q) $ pretzel knots where $p_i,q>0$ and $q = \min(p_1,\dots,p_n)$.