Did you mean $G\wr S_n\cong G^n\rtimes S_n$?

Thank you, Fedor! I knew that the answer was in there, but did not have time this morning to find the exact place. I have updated the answer.

If a representation factors through a finite quotient, then you can lift it to the Weil group, but you can also lift it to the full Galois group. Moreover, the former lift will be the restriction of the latter. So I guess I showed that the whole lift to the full Galois group will already be determined by the local polynomial. But really, that distinction is empty for representations that factor through a finite quotient.

No, you were not using bad notation, you had incorrect statements. It is now fixed.

I don't know anything about logic, and maybe I am missing something here, but unless you are using some non-standard notation, your characterisation of $[F:K]$ is wrong. It does not apply to any field extension of degree $3$, for example. The claim about $\lambda < \kappa$ is, accordingly, also wrong for degree $3$ extensions.

Very nice, Filippo! I am very happy that this question persuaded you to polish that paper and make it public; that effect alone made this question worthwhile!

Thanks, Derek, I could not remember what it was called.