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
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 $\bar Q\_p$.


