2 added 81 characters in body

Let me give at least a partial answer.

If you want to view the 1 and 2 dimensional theories uniformly, you should look at everything adelically. In dimension 1, Dirichlet characters can be viewed as idele class (or Hecke) characters, which is to say irreducible representations of $\mathbb Q^\times \backslash \mathbb A^\times$. Hence 1-dimensional Galois representations correspond to automorphic representations of $GL_1(\mathbb A)$. (Edit: I forgot to mention, this correspondence is abelian class field theory!)

In general, $n$-dimensional Galois representations conjecturally correspond to automorphic representations of $GL_n(\mathbb A)$. There's a standard passage from modular forms to (classical then adelic) automorphic forms, and from automorphic forms to automorphic representations of $GL_2(\mathbb A)$. The "odd" 2-dimensional Galois representations should correspond to automorphic representations coming from modular forms. (Incidentally, to get the Euler product for the L-function of a modular form, it should be an eigenform.) Interpreted in terms of automorphic representations, one sees that (the L-functions for) modular forms are a higher dimensional analogue of (the L-functions for) Dirichlet characters.

To change the base field, work with representations of $Gal(\bar K/K)$ on the Galois side, and $GL_n(\mathbb A_K)$ on the automorphic side.

For the last question, if $K/\mathbb Q$ is Galois, then yes, you have the same type of factorization for the Dedekind zeta function (realize it as an Artin L-function for the trivial representation).

There are several places you can read survey articles. For instance, the book "Introduction to the Langlands Program." Knapp also has nice survey articles, and Gelbart has a nice Bulletin article. I have links to some articles here.

If you want to view the 1 and 2 dimensional theories uniformly, you should look at everything adelically. In dimension 1, Dirichlet characters can be viewed as idele class (or Hecke) characters, which is to say irreducible representations of $\mathbb Q^\times \backslash \mathbb A^\times$. Hence 1-dimensional Galois representations correspond to automorphic representations of $GL_1(\mathbb A)$.
In general, $n$-dimensional Galois representations conjecturally correspond to automorphic representations of $GL_n(\mathbb A)$. There's a standard passage from modular forms to (classical then adelic) automorphic forms, and from automorphic forms to automorphic representations of $GL_2(\mathbb A)$. The "odd" 2-dimensional Galois representations should correspond to automorphic representations coming from modular forms. (Incidentally, to get the Euler product for the L-function of a modular form, it should be an eigenform.) Interpreted in terms of automorphic representations, one sees that (the L-functions for) modular forms are a higher dimensional analogue of (the L-functions for) Dirichlet characters.
To change the base field, work with representations of $Gal(\bar K/K)$ on the Galois side, and $GL_n(\mathbb A_K)$ on the automorphic side.
For the last question, if $K/\mathbb Q$ is Galois, then yes, you have the same type of factorization for the Dedekind zeta function (realize it as an Artin L-function for the trivial representation).