You might find the Coleman--Edixhoven paper "On the semisimplicity of the Up operator" illuminating (the title is about the Hecke operator U(p) on modular forms, but the proofs proceed by relating the problem to semisimplicity of Frobenius on DdR).

I don't understand: how does computing in $SL_2(\mathbb{Z} / 12)$ show that no further relations are needed to compute the abelianisation for $n \ne 12$?

OK, I see now that you are not claiming this, but then how is the assertion proved?

I'm a little surprised by the assertion about "the only extra relation you need for $SL_2(\mathbb{Z}/n)$ is...". I've seen presentations in the literature for $SL_2(\mathbb{Z}/n)$ in terms of $U, V$, and even used them in my own work, but they've always been substantially more complicated than that one -- see e.g. page 2 of arxiv.org/pdf/1307.0625.pdf. That doesn't per se mean that your statement is false, but it makes me curious to see a proof.

Have you tried contacting the authors?

It seems that, after Peter Humphries had given a short but informative answer to the original question in the comments, the original post was edited to add a second (only very distantly related) question onto the end of the original one. If you have something new to ask, then please ask a new question.

Yes, I think it is still true. Try to prove it for an Eisenstein eigenfrom (where the Galois rep is isomorphic to the direct sum of two characters).

It's correct when the tensor product of K and E (over Qp) is a field , ie if the Galois closures of K and E are are linearly disjoint. This includes the case when either K or E is Qp.

Clearly it does count as a "standard reference" but I claim it does not count as "making explicit", given that Borel's proof is very brief indeed. :-)

Since Mikhail has posted this reference to his work again, I am posting again the comment that this work does not actually involve $\mathbb{Z}_p$-extensions in the sense of the question (i.e. Galois extensions with Galois group isomorphic to $\mathbb{Z}_p$).

I have never heard of this "Serre twisting" operation before (and I have not attempted to check that your formula gives a well-defined representation, which is not obvious to me); but since the weight of a Galois representation is a condition on the eigenvalues of Frobenius elements, and conjugation doesn't change the eigenvalues, it is clear that if ${}_c V$ exists then it will be pure of the same weight as $V$.

Please don't ask new questions in comments.