Looks like a famous (and famously hard) open problem to me.

@AlexeyUstinov The epigraph is a quotation from a book by French sociologist Pierre Bourdieu reading "...qui a le loisir de s’arracher aux évidences de l’existence ordinaire pour se poser des questions extra-ordinaires ou pour poser de manière extra-ordinaire des questions ordinaires". As ACL writes above, it might be necessary to read the acknowledgement section and the title or introduction of the thesis to get why I chose it.

@ მამუკა ჯიბლაძე Yes, precisely. I find it interesting that, contrary to the common apocrypha, the actual quote makes the role and even form of mathematics precise and that, far from being theistic, it also makes it clear that it is because we are human that mathematics is required.

The Galois representation attached to a twist of a modular form is the twist of the Galois representation (more generally, that is a big part of the internal compatibility required of the Global Langlands Correspondence). This answers all your questions and more.

@Bonbon It depends what you want to do. If you want to retrace the history of the subject, many other references can (and should) be mentioned. But if your aim is to understand a proof of the theorem, then you can read Falting's article (proving $\rho_f|G_{\mathbb Q_p}$ is Hodge-Tate), then Tsuji's (proving it is potentially semi-stable hence de Rham) and finally Saito's (proving the Weil-Deligne representation is the one attached to $\pi(f)_p$ by the Local Langlands Correspondence.

Based on the comments in MR1279611, I think it is reasonable to infer that Carayol was the first to publish the result but that Serre was aware of it in the mid-1980s. However, I doubt he communicated it explicitly before becoming aware of Carayol's result, as Barry Mazur, for instance, seems to be unaware of it in Deforming Galois Representations, which is from 1989.

I think that part of the difficulty is that there might be a typo in Mazur's statement: for the definition to be coherent, I think $(R^q\tau M,0,0)$ should be $(0,R^q\tau M,0)$ (otherwise his statement do not apply already when $q=0$).