Some questions on the $p$-adic properties of special $L$-values Warning: Some naive, speculative questions from a total non expert.
Let $\rho$ be a p-adic representation of the Galois group $Gal(\overline K/K)$ for a number field $K$. We can consider the Artin L-function $L(s,\rho)$ associated to the representation. Let me define $L_p(s,\rho)$ to be $L(s,\rho)$ with the Euler factor at (primes above) $p$ removed. 
Now, in certain special cases we have congruences of the form:
$$L_p(1-n,\rho_1) \equiv L_p(1-n,\rho_2)\pmod{\pi}$$
for $n \geq 1$ if $\rho_1 \equiv \rho_2 \pmod{\pi}$. This congruence might not even make sense in the general case (what is the reduction of a transcendental number?) but at least for, say, Dirichlet characters, these are just a form of the Kummer congruences (if we place some restrictions on what $\rho_1,\rho_2$ can be).
We can think of this congruence as indicating that something like the following must be true:
"It is possible to define a mod p Artin function $\overline L(s,\overline\rho)$ that depends only on the reduction $\overline\rho$ of the reduction."
Of course this is very ambitious and very ill defined in general. Nevertheless, my questions are as follows:
1) What generalizations of the Kummer congruences are known? This is probably the same question as asking "what definitions of p-adic L functions are known?". I know that it is possible to prove such results for totally real number fields but do we know of any other cases? I am interested in general L-functions here, not just Artin L-functions.
2) Have there been any attempts to actually define a mod p Artin function of the kind indicated above? Even in very simple cases (say for the Riemann zeta function), I have not come across any indication of such an attempt (which is not saying much).
A naive idea might be to just take a power series representation of a p-adic L function and reduce each term formally. For instance, let $\chi$ be a Dirichlet character, $L_p(s,\chi) = a_{-1}/(s-1) + a_0 + a_1(s-1) + \dots$ be the corresponding p-adic L-series with $a_n \in \mathbb Q_p(\chi)$. However, for this to even have a hope of working, the coefficients $a_n$ of the Taylor series should be p-adic integers instead of just arbitrary p-adic numbers. I suspect this is false...
 A: You might enjoy reading this paper: Avner Ash and Glenn Stevens, "Modular forms in characteristic $\ell$ and special values of their L-functions", Duke Math. J. 53 (1986), no. 3, 849-868. The gist of this paper is that (in certain nice cases) congruences between modular forms give rise to congruences between their L-functions.
In general, one reason why this problem is hard is because $L(s, \rho)$ will very rarely be algebraic without normalising by some period first. (The example of 1-dimensional $\rho$'s is perhaps a bit misleading here -- there are very few interesting examples in higher dimension where no period is needed.) So $L(s, \rho)$ depends not only on $\rho$ but also on the period you choose, and if you want to see congruences, you need to choose periods "compatibly" for $\rho_1$ and $\rho_2$. 
So at the moment,we only have results when $\rho$ comes from some specific kind of representation (e.g. from modular forms) which allows you to understand the periods involved. In the general case there are lots of conjectures (e.g. Fukaya and Kato have a very general formulation) but no theorems!
A: 
1) What generalizations of the Kummer congruences are known?

This is somewhat imprecise as a question and in particular, I would dispute a little your assertion that

This is probably the same question as asking "what definitions of $p$-adic L functions are known?"

because the existence of a $p$-adic $L$-function is a much stronger result than the existence of congruences of the type you asserted or even of the mod $p$ $L$-function.
To give a concrete example, there is a mod $3$ $L$-function attached to the common residual representation of the two elliptic curves $$E_1:y^2=x^3-7x-6$$
and
$$E_2:y^2=x^3-27x+486$$
but in the current understanding of the term, I don't think it is even meaningful to ask whether there is a $p$-adic $L$-function $p$-adically interpolating the special values of the attached modular forms (if I'm not mistaken, the modular form attached to the first one has finite slope at $3$ but not the modular form attached to the second one, so these two forms do not belong to a family for which we know a $p$-adic $L$-function).
As you noted already in your question and as was again emphasized by David Loeffler, one of the difficulty is in the choice of periods, or equivalently in the pin-pointing of which special values exactly you intend to have enter the congruence. Note also that the formulation you propose

define a mod $p$ Artin function $\bar{L}_{p}(\bar{\rho},s)$

seems not only to presupposes that $L_{p}(\rho,s)$ is algebraic but also that its $p$-adic valuation is positive, and that need not be true in general.
Finally, even when all these problems are solved, there is the little problem that congruent Galois representations need not have congruent special values, as was first observed I think by R.Greenberg and V.Vatsal for the pair of elliptic curves congruent modulo 5
$$E_3:y^2=x^3+x-10$$
and
$$E_4:y^2=x^3-584x+5444$$
(the special value $L(E_3,1)$ divided by the appropriate period is a 5-adic unit while it is not the case of the second). At first sight then, there can be no naive $\bar{L}_{5}(\bar{\rho},s)$ satisfying your desiderata for $\bar{\rho}$ the residual representation $E_3[5]$ (in a way, this phenomenon already appears for characters and lies lurking in your observation that one has to "place some restrictions on what $\rho_1$ and $\rho_2$ can be").
What can be done, following Kato and Fukaya-Kato, is to define a mod $p$ object $\mathcal L(\rho)$ which - if the conjecture on special values are true - encodes whether or not the special values divided by the period specified in the Bloch-Kato conjectures is a $p$-adic unit or not. More precisely, that element $\mathcal L(\rho)$ belongs to $\mathbb F_p$ and is well-defined modula an element of $\mathbb F^\times_p$ (sic!). The $\mathcal L(\rho)$ is a rather weak version of a mod $p$ $p$-adic $L$-function but probably the best you can do in that level of generality in view of the multiple ambiguities in the definition of special values to begin with.
Independently of the truth of the conjecture on special values, one may wonder if $\mathcal L(\rho_1)$ is equal to $\mathcal L(\rho_2)$ whenever $\rho_1\equiv\rho_2$ modulo $p$. If that is the case, we may denote this common element $\mathcal L(\bar{\rho})$ (remember though that $\mathcal L(\bar{\rho})$ is just an element of $\mathbb F_p$ modulo $\mathbb F^\times_p$). In general, one cannot even do this, as the pair $E_3,E_4$ above establishes.
However, if either 1) we strip $L$-functions of their Euler factors at all primes of bad reduction (that case is mostly formal) or 2) the two Galois representations are geometric* and belong to the same irreducible components of the universal deformation space of their common $\bar{\rho}$ (that case requires some work), then $\mathcal L(\bar{\rho})$ is well-defined. One can find counter-examples showing that the requirement that both representations be geometric is necessary (that is to say, there exist congruent Galois representations belonging to the same irreducible component of the universal deformation space of their common $\bar{\rho}$ but with different $\mathcal L(\rho)$).
In conclusion, yes, congruences between special values coming from congruences between the underlying objects (Kummer congruences, if you want) abound and there is an object $\mathcal L(\bar{\rho})$ a bit like what you wish for, but that object carries very little information, is known to exist only under specific hypotheses on the $\rho_i$ and typically requires the previous knowledge of strong versions of the conjectures on special values of $L$-functions to be really meaningful.        
*To be precise, this requires a rather strong understanding of geometric, in which the Weight-Monodromy Conjecture is supposed to hold for geometric Galois representations.
A: There is a modification of the quantity $\mathcal{L}(\rho_E)$ attached to the Galois representation of an elliptic curve $E$, for any Artin representation $\tau:G_{\mathbb{Q}}\rightarrow GL_n(\bar{\mathbb{Q}})$ there is the "twisted" version which has been considered for instance by T Dokchitser and V Dokchitser (with appendix by Coates and Sujatha) Computations in Noncommutative Iwasawa Theory. The collection of all values $\{\mathcal{L}_E(\tau)\}$ as $\tau$ ranges over all Artin representations captures more information about $E$ than the single value $\mathcal{L}(\rho_E)$.
Let $m$ be a postive integer coprime to $p$ and let $F_{\infty}$ be the extension of $\mathbb{Q}(\mu_{p^{\infty}})$ obtained by adjoining all $p$ power roots of $m$. Let $G=\text{Gal}(F_{\infty}/\mathbb{Q})$ and $\Lambda(G)$ its Iwasawa algebra. For self-dual Artin representation $\tau$ which factors through $G$, the Main conjecture asserts that the order of $p$ which divides $\mathcal{L}_{E}(\tau)$  should equal the order of $p$ which divides the $G$-Euler-characteristic of the dual Selmer group of $E[p^{\infty}]$ over the non-abelian extension $F_{\infty}$. This is the context in the above paper.
Expecting that $\mathcal{L}_E(\tau)\equiv \mathcal{L}_{E'}(\tau)\mod{p}$ for congruent elliptic curves (with lots of hypotheses) would perhaps better approximate the expectation you have in mind. Perhaps one should restrict oneself to twists by characters only, but these non-abelian twists have also been considered.
