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This is not such a contrast to Mazur's setup as you seem to think, since $\mathbb{Z}_p[[X]]$ embeds in the ring you are calling $\mathbb{Z}_p\langle X / r\rangle$ for any $r < 1$.
Incidentally, there are computer programs for this sort of thing: given any elliptic curve over Q, Sage or Magma will happily compute for you a list of all the curves isogenous to it.
Unless you know more about your representation this question is unanswerable. It’s trivial to check that if T is a Zp-lattice in a crystalline representation, then you can always find T’ which is non-crystalline and congruent to T modulo an arbitrarily high power of p.
FWIW, the notation R<X/r> already has a meaning and it isn’t that. But the notational question is unimportant. I think @tkr has very correctly diagnosed the misconception behind the question.
I agree with the substance of Will’s comment - the question isn’t answerable until you tell us in what way the V in your problem is characterised. That said, a single element of Qp is in itself an “infinite amount of data”!
The $p$-adic, $T$-adic, and $(p, T)$-adic topologies on $\Lambda$ are all different. For most purposes it is the $(p, T)$-adic one which is important (since it is the compact one); the other two topologies are less useful.
Sure, he computes with the $gl_2$ adjoint and gets a module with two summands, one from $k$ and one from $sl_2$ – both of which are apparently non-trivial when $p = 3$.
In that case I am puzzled. Herzig seems to be saying that $H^1(SL_2(\mathbb{F}_3), \mathrm{ad}^0) = 0$. But the first line of the proof of Lemma 3.3 in Bell's paper doi.org/10.1016/0021-8693(78)90027-3 seems to show that, for any $p$ and any module $M$ of $p$-power order, we have $H^1(SL_2(\mathbf{F}_p), M) = H^1(U, M)^T$ where $U$ and $T$ are the usual unipotent and torus subgroups. Lemma 1.2 of Flach (eudml.org/doc/144023) computes $H^1(U, ad^0)$ as a $T$-module and shows that it is 1-dimensional with $diag(t, t^{-1}) \in T$ acting as $t^4$.
I looked up Hertzig's paper but it is exceedingly terse and the notations are not familiar to me. He defines two groups $\Gamma$ and $\Delta$ for each root system. Can @testaccount, or some other kind person, explain to me what these are for $A_1$? Presumably one of them is $SL_2(k)$ but it's not clear to me which.
If I understand CPS correctly, they deal with all finite fields of characteristic $\ne 2$ with $q > 5$. In my application the char. 2 case is banned for other reasons anyway, so the problem is only the prime fields $\mathbb{F}_3$ and $\mathbb{F}_5$. Now I need to sit down and work out how $k^\times$ acts on $H^1(SL_2(k), \mathrm{ad})$ in these cases...
