A theorem of Gouvea and Hida (or rather a consequence of it) states that there exist a Galois representation attached to a $p$-adic eigenform $f$ provided the residual representation attached to a classical eigenform $g$ congruent to $f$ is absolutely irreducible. I believe there might be generalizations due to Skinner-Wiles. So my question is: do there always exist a Galois representation attached to an ordinary $p$-adic modular form? What if $g$ has some extra properties like CM but I still want the residual representation to be reducible?
Representations Attached to p-adic Modular Forms – Number Theory
galois-representationsmodular-formsnt.number-theory
Related Solutions
The right way to do this sort of question is to apply Saito's local-global theorem, which says that the (semisimplification of the) Weil-Deligne representation built from $D_{pst}(\rho_{f,p})$ by forgetting the filtration is precisely the one attached to $\pi_p$, the representation of $GL_2(\mathbf{Q}_p)$ attached to the form via local Langlands. Your suggestions about the $p$-adic valuation of $N$ and so on are rather "coarse" invariants---$\pi_p$ tells you everything and is the invariant you really need to study.
So now you can just list everything that's going on. If $\pi_p$ is principal series, then $\rho$ will become crystalline after an abelian extension---the one killing the ramification of the characters involved in the principal series. If $\pi_p$ is a twist of Steinberg by a character, $\rho_{f,p}$ will become semistable non-crystalline after you've made an abelian extension making the character unramified. And if $\pi_p$ is supercuspidal, $\rho_{f,p}$ will become crystalline after a finite non-trivial extension that could be either abelian or non-abelian, and figuring out which is a question about $\pi_p$ (it will be a base change from a quadratic extension if $p>2$ and you have to bash out the possibilities).
Seems to me then that semistable $\rho$s will show up precisely when $\pi_p$ is either unramified principal series or Steinberg, so the answer to your question is (if I've got everything right) that $\rho_{f,p}$ will be semistable iff either $N$ (the level of the newform) is prime to $p$, or $p$ divides $N$ exactly once and the component at $p$ of the character of $f$ is trivial. Any other observations you need should also be readable from this sort of data in the same way.
One consequence of this I guess is that $\rho_{f,p}$ is semi-stable iff the $\ell$-adic representation attached to $f$ is semistable at $p$.
Given an eigencuspform $f$ of weight $k≥2$ (so in particular 2) and $p$ a prime of ordinary reduction (in particular good ordinary reduction), there is always a Hida family passing through $f$. This is for instance Theorem I of Galois representations into $\operatorname{GL}_{2}(\mathbb Z_{p}[[X]])$ attached to ordinary cusp forms by H.Hida (Inventiones Mathematicae, 1986).
As Noam Elkies writes in comment, the second question appears at present to be hopelessly hard. It is generally believed that all examples of elliptic curves congruent modulo a large prime $p$ should be very restricted but I don't think we are anywhere near a proof.
In summary, the answer to your first question is the best possible (positive, with well-documented references) while the answer to your second question is the worse possible (probably negative but nobody knows).
Best Answer
I am not sure this answer will satisfy you totally because I am not sure what you mean exactly by $p$-adic modular forms. But at least, for a $p$-adic modular form $f$ in the sense of Serre, which is an eigenform for almost all the Hecke operators $T_\ell$ (with eigenvalue $a_\ell)$ there always exists a semi-simple Galois representation $r:G_{\mathbb Q} \rightarrow Gl_2( \bar Q_p)$ such that $tr(\rm Frob_\ell)=a_\ell$ for almost all $\ell$. This is quite easy to prove with modern techniques which were not available at the time Serre, Hida, and Gouvêa worked on the subject. Here is how:
Lemma : for every classical normalized form $g$ of weight $k$ and level $\Gamma$ with coefficient in $\mathbb Z_p$ such that $T_\ell g \equiv a_\ell g \pmod{p^n}$ for almost all $\ell$, there exists a unique continuous pseudo-character $T_g : G_{\mathbb Q} \rightarrow \mathbb{Z}/p^n \mathbb{Z}$ such that $T_g(\rm Frob_\ell) = a_\ell$ in $\mathbb Z/p^n \mathbb Z$
Proof: consider the Hecke algebra $A$ generated by the Hecke operators $T_\ell$ acting on $M_k(\Gamma,\mathbb Z_p)$ (the module of modular forms of weight $k$, level $\Gamma$, coefficients in $\mathbb Z_p$). There is pseudo-character $T : G_Q \rightarrow A$ such that $T(Frob_\ell)=T_\ell$. To prove this, first prove it with $A$ replaced by $A \otimes_{\mathbb Z_p} K$ where $K$ is is some sufficiently large extension of $\mathbb Q_p$. Then $A \otimes_{\mathbb Z_p} K = K^d$ where each factor corresponds to an eigenform in $M_k(\Gamma,K)$. Hence to construct $T : G \rightarrow K^d$, one just takes the sum of the trace of the representations attached to these eigenforms. Then $T(\rm Frob_\ell)=T_\ell$ by construction, so $T$ takes values in $A$ on a dense subset of $G_{\mathbb Q}$, so $T$ takes values in $A$ everywhere since $A$ is closed in $A \otimes_{Z_p} K$. To finish the proof of the existence part of the lemma, just compose $T$ with the morphism of ring $A \rightarrow \mathbb Z/p^n \mathbb Z$ which sends $T_\ell$ on $a_\ell$. The uniqueness is clear by Cebotarev.
Now, back to the main claim, for every $n$, $f$ is by definition congruent to a classical form $g$ mod $p^n$, hence by the lemma we get a pseudo-character $T_n: G_{\mathbb Q} \rightarrow \mathbb Z/p^n \mathbb Z$ such that $T_n(\rm Frob_\ell)$ $ = a_\ell \pmod{p^n}$. By uniqueness, we can glue those pseudo-characters into
a pseudo-character $T_\infty: G_{\mathbb Q} \rightarrow \mathbb Z_p \rightarrow \mathbb Q_p$ and by a theorem of Taylor, this pseudo-character is the trace of a representation over $\bar Q_p$.