Let $K$ be an algebraically closed field. The finite-dimensional representation theory of the polynomial algebra $K[x]$ is tame and completely understood, which I shall first summarise. It's indecomposable modules are determined by Jordan blocks
$$ J_{\lambda,m}= \begin{pmatrix}
\lambda & 1 & 0 & \cdots & 0 \\
0 & \lambda & 1 & \cdots & 0 \\
\vdots & \vdots & \ddots & \ddots & \vdots \\
0 & 0 & \cdots & \lambda & 1 \\
0 & 0 & \cdots & 0 & \lambda
\end{pmatrix}.$$
The simple modules are precisely the 1-dimensional modules. For every simple module $M_1$ defined such that the action of $x$ on $M_1$ is scalar multiplication by $\lambda \in K$, we have an infinite chain of submodules
$$ M_1 \subset M_2 \subset M_3 \subset \ldots,$$
where the action of $x$ on $M_i$ is a linear transformation by $J_{\lambda, i}$. In particular, such a chain of indecomposables induces a homogeneous tube in the Auslander-Reiten quiver $\Gamma$ of finite-dimensional modules of $K[x]$, and all components of $\Gamma$ are homogeneous tubes with a simple module at the mouth. As a consequence, there is an irreducible monomorphism $M_i \rightarrow M_{i+1}$ and irreducible epimorphism $M_{i+1} \rightarrow M_i$ for all $i$ in the chain above. If

$$ M'_1 \subset M'_2 \subset M'_3 \subset \ldots,$$
is a different chain (so $M_1 \not\cong M'_1$), then $\mathrm{Hom}_{K[x]}(M_i, M'_j)=0$ and $\mathrm{Hom}_{K[x]}(M'_i, M_j)=0$ for any $i$ and $j$.

**My questions consist of two parts:**

(1) What is known about the finitely-generated (but not necessarily finite-dimensional) representation theory of $K[x]$? Is the category of finitely-generated representations tame (i.e. is it possible to classify the indecomposable representations and/or $\mathrm{Hom}$-spaces? An explanation/summary accompanied by references would be greatly appreciated (if this is feasible, that is).

(2) Is anything at all known about the infinite-dimensional (but not necessarily finitely-generated) representation theory of $K[x]$? My instinct says that this category is likely wild (but a confirmation/rebuttal is welcome). Even so, are there known examples or families of infinitely-generated representations that could be considered interesting, or could have potential applications to other areas of mathematics (such as mathematical physics)?