@user471019 Somewhat unfortunately, I believe the answer to the question "what do I need to know about $p$-adic Hodge theory to understand Saito's result ?" is basically "all of it": when the weight of the modular form is large compared to $p$, I don't think there is any significant short cut to obtain Ccrys (the statement that good reduction implies crystalline) just for the Kuga-Sato variety (rather than for a general propre smooth scheme with good reduction). On the other hand, you can certainly understand what Saito does and what it means while taking the proof of Ccrys as a black box.

@user471019 I think the first complete published proof removing the condition for large $p$ and independent of Faltings's work (which does contain a gap), is in Tsuji, Inventiones 1999 (not that this paper was available already to Saito in 1997 and that Saito's paper crucially relies on it, even to state its main theorem). Falting's proof was completed in 2002 (Almost étale extensions) but most human beings find it hard to digest nevertheless.

As $\mathcal O_{K}$ is a DVR, the statements and proofs are exactly the same.

@DavidCorwin Yes (the latter verification being usually easy for small primes independently of the algebraic rank of the elliptic curves because it is enough to bound the order of vanishing of the p-adic L-function).

Indeed, $p=163$ is totally ramified, and $p=5,13$ or 17 split completely in 3 non-principal ideals.

@GHfromMO Yes, of course but Jean wants degree exactly $n$, I believe, and though I thought it would follow easily, I could not find a convincing argument.